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color: var( --color-emphasized, #000000 ); font-size: 95%; padding: 0.2em 0.2em 0.2em 2em; margin-bottom: 1em; border: 1px solid #b6b6b6;"><i>Tämä artikkeli käsittelee laskutoimitusta. Nimitystä potenssi käytetään myös seksuaalisesta kyvykkyydestä, katso artikkeli <a href="/wiki/Impotenssi" title="Impotenssi">Impotenssi</a>.</i></div> <p><b>Potenssi</b> on <a href="/wiki/Matematiikka" title="Matematiikka">matemaattinen</a> lyhennysmerkintä, jolla esitetään saman luvun toistuva <a href="/wiki/Kertolasku" title="Kertolasku">kertolasku</a>. Esimerkiksi kolmen 2:n tulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot 2\cdot 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot 2\cdot 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbb690bd039a88481d0bf532f475c760e272321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.846ex; height:2.176ex;" alt="{\displaystyle 2\cdot 2\cdot 2}"></span> lyhennetään <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e9f8299773e9205d2055998f3c8eb9441877fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{3}}"></span>. Toistuvaa lukua kutsutaan <i>kantaluvuksi</i> ja toiston lukumäärää <i>eksponentiksi</i>, jolloin merkinnässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e9f8299773e9205d2055998f3c8eb9441877fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{3}}"></span> luku 2 on kantaluku ja luku 3 on eksponentti. Tällöin sanotaan, että luku 2 <i>korotetaan potenssiin</i> 3. Arkipäiväisemmin sanotaan myös "kaksi <b>potenssiin</b> kolme", "kaksi kolmanteen <b>potenssiin</b>" tai lyhyemmin "kaksi kolmanteen". </p><p>Yleisesti voidaan merkitä kantaluvun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> korottamista potenssiin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot a\cdot ...\cdot a=a^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot a\cdot ...\cdot a=a^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce634779b8c118b46447fb367df0b15a9461ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.311ex; height:2.343ex;" alt="{\displaystyle a\cdot a\cdot ...\cdot a=a^{n}}"></span>. Merkintää voidaan lukea myös "a <b>potenssiin</b> n", "a n:nteen <b>potenssiin</b>" tai "a:n n:s <b>potenssi</b>". <sup id="cite_ref-r1_1-0" class="reference"><a href="#cite_note-r1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="fi" dir="ltr"><h2 id="mw-toc-heading">Sisällys</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Käsitteitä_ja_merkintätapoja"><span class="tocnumber">1</span> <span class="toctext">Käsitteitä ja merkintätapoja</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Potenssin_laskemisesta"><span class="tocnumber">2</span> <span class="toctext">Potenssin laskemisesta</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Ominaisuudet"><span class="tocnumber">2.1</span> <span class="toctext">Ominaisuudet</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Eksponenttina_positiivinen_kokonaisluku"><span class="tocnumber">2.2</span> <span class="toctext">Eksponenttina positiivinen kokonaisluku</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Eksponenttina_nolla"><span class="tocnumber">2.3</span> <span class="toctext">Eksponenttina nolla</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Negatiivinen_eksponentti"><span class="tocnumber">2.4</span> <span class="toctext">Negatiivinen eksponentti</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Eksponenttina_rationaaliluku"><span class="tocnumber">2.5</span> <span class="toctext">Eksponenttina rationaaliluku</span></a></li> <li class="toclevel-2 tocsection-8"><a href="#Miksi_kantaluvun_on_oltava_positiivinen?"><span class="tocnumber">2.6</span> <span class="toctext">Miksi kantaluvun on oltava positiivinen?</span></a></li> <li class="toclevel-2 tocsection-9"><a href="#Eksponenttina_irrationaalinen_luku"><span class="tocnumber">2.7</span> <span class="toctext">Eksponenttina irrationaalinen luku</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#Potenssiin_perustuvia_funktioita"><span class="tocnumber">3</span> <span class="toctext">Potenssiin perustuvia funktioita</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Katso_myös"><span class="tocnumber">4</span> <span class="toctext">Katso myös</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Lähteet"><span class="tocnumber">5</span> <span class="toctext">Lähteet</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Käsitteitä_ja_merkintätapoja"><span id="K.C3.A4sitteit.C3.A4_ja_merkint.C3.A4tapoja"></span>Käsitteitä ja merkintätapoja</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=1" title="Muokkaa osiota Käsitteitä ja merkintätapoja" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=1" title="Muokkaa osion lähdekoodia: Käsitteitä ja merkintätapoja"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Luvun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> toista potenssia eli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f564e5dc0b6e68af32ca8614e972f5b36e944a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.676ex;" alt="{\displaystyle a^{2}}"></span> kutsutaan usein luvun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> <i>neliöksi</i> ja vastaava kolmatta potenssia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> <i>kuutioksi.</i> Siten merkintä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3509ed003dec434ccbed30b858850e928ed70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 4^{2}}"></span> voidaan lausua "luvun neljä neliö" eli "neljän neliö" ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db2df10fa3880f4d05e7721fbe33324c0e0ee12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 4^{3}}"></span> "luvun neljä kuutio" eli "neljän kuutio". </p><p>Erityisesti laskimissa käytetään luvun <a href="/wiki/Kymmenpotenssimuoto" title="Kymmenpotenssimuoto">kymmenen potensseille</a> erityistä merkintäänsä. Esimerkiksi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11abe9466792939e33072aaed72a029e9389030a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{2}}"></span> merkitään 1E+2, joka tarkoittaa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot 10^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 10^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a0459408db3f734b57c498ee1c5f3a782c4eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.221ex; height:2.676ex;" alt="{\displaystyle 1\cdot 10^{2}}"></span>. Luku 1 on siis kerroin, kirjain E ilmoittaa, että on kyse kymmenen potensseista, ja +2 tarkoittaa kymmenen positiivista eksponenttia kaksi. Vastaavasti merkittäisiin esimerkiksi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{,}3\cdot 10^{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>3</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{,}3\cdot 10^{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1001d43f5f1653f18163ddedee87c7daa188fd62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.03ex; height:3.009ex;" alt="{\displaystyle 2{,}3\cdot 10^{6}}"></span> muodossa 2,3E+6. </p> <div class="mw-heading mw-heading2"><h2 id="Potenssin_laskemisesta">Potenssin laskemisesta</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=2" title="Muokkaa osiota Potenssin laskemisesta" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=2" title="Muokkaa osion lähdekoodia: Potenssin laskemisesta"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ominaisuudet">Ominaisuudet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=3" title="Muokkaa osiota Ominaisuudet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=3" title="Muokkaa osion lähdekoodia: Ominaisuudet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Potenssi ei ole <a href="/wiki/Vaihdannaisuus" title="Vaihdannaisuus">vaihdannainen</a> kuten yhteen- tai kertolasku. Esimerkiksi, <span style="white-space:nowrap">2 + 3 = 3 + 2 = 5</span> ja <span style="white-space:nowrap">2 · 3 = 3 · 2 = 6</span>, mutta <span style="white-space:nowrap">2<sup>3</sup> = 8</span>, kun taas <span style="white-space:nowrap">3<sup>2</sup> = 9</span>. </p><p>Potenssi ei ole myöskään <a href="/wiki/Liit%C3%A4nn%C3%A4isyys" title="Liitännäisyys">liitännäinen</a>. Esimerkiksi <span style="white-space:nowrap">(2 + 3) + 4 = 2 + (3 + 4) = 9</span> ja <span style="white-space:nowrap">(2 · 3) · 4 = 2 · (3 · 4) = 24</span>, mutta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{3})^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2^{3})^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30b41a6a23eda501f1ebdb2cb15f9aa70e5569c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.08ex; height:3.176ex;" alt="{\displaystyle (2^{3})^{4}}"></span> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbc46044bf4df8cf5792d5886cc9877306d6ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 8^{4}}"></span> = 4 096, kun taas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{(3^{4})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{(3^{4})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/614522aa2edec0f6daadfb2858f366958409260b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.328ex; height:3.009ex;" alt="{\displaystyle 2^{(3^{4})}}"></span> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{81}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>81</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{81}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b111c87b1393356316b5530e119dadcb9a93859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.039ex; height:2.676ex;" alt="{\displaystyle 2^{81}}"></span> = 2 417 851 639 229 258 349 412 352. </p><p>Jos sulkeita ei ole merkitty, lasketaan potenssit alkaen ylimmästä eksponentista: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{p^{q}}=b^{(p^{q})}\neq (b^{p})^{q}=b^{(p\cdot q)}=b^{p\cdot q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≠</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{p^{q}}=b^{(p^{q})}\neq (b^{p})^{q}=b^{(p\cdot q)}=b^{p\cdot q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b72ae758648eddd3402c15319a2296e82dfdc8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.018ex; height:3.343ex;" alt="{\displaystyle b^{p^{q}}=b^{(p^{q})}\neq (b^{p})^{q}=b^{(p\cdot q)}=b^{p\cdot q}}"></span> <br />(Kirjoitettuna kaavana: <i>b^p^q = b^(p^q) ≠ (b^p)^q</i>.)</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Eksponenttina_positiivinen_kokonaisluku">Eksponenttina positiivinen kokonaisluku</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=4" title="Muokkaa osiota Eksponenttina positiivinen kokonaisluku" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=4" title="Muokkaa osion lähdekoodia: Eksponenttina positiivinen kokonaisluku"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Edellä esitetty potenssin havainnollinen tulkinta voidaan kirjoittaa muodollisesti seuraavasti. Olkoon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> reaaliluku ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> <a href="/wiki/Positiivinen_luku" title="Positiivinen luku">positiivinen</a> <a href="/wiki/Kokonaisluku" title="Kokonaisluku">kokonaisluku</a>. Tällöin määritellään <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{1}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{1}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7545d7ed2d371323fca6aa910f4e9a41f86cd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.612ex; height:2.676ex;" alt="{\displaystyle a^{1}=a}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}=a\cdot a^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}=a\cdot a^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77e4b591263130f5cfb392dda5f955dfa141a1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.004ex; height:2.676ex;" alt="{\displaystyle a^{n}=a\cdot a^{n-1}}"></span>, kun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></span>. </p><p>Tulon tekijöiden lukumääriä tarkastelemalla voidaan todistaa seuraavat laskusäännöt päteviksi, kun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> ovat reaalilukuja sekä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> positiivisia kokonaislukuja: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{m}a^{n}=a^{m+n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{m}a^{n}=a^{m+n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09f2da5096059eabac7554aab77c2257acc7a60a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:14.008ex; height:2.509ex;" alt="{\displaystyle a^{m}a^{n}=a^{m+n}\,\!}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}\qquad (a\neq 0,m>n)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>m</mi> <mo>></mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}\qquad (a\neq 0,m>n)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d11a2b7791e63d47af3caffb8ce0d87e77c622ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:31.209ex; height:3.676ex;" alt="{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}\qquad (a\neq 0,m>n)\,\!}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{m})^{n}=a^{mn}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{m})^{n}=a^{mn}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76f6545f9d4aa068e076c3369dd78cbc839e2b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:13.309ex; height:2.843ex;" alt="{\displaystyle (a^{m})^{n}=a^{mn}\,\!}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}b^{n}=(ab)^{n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}b^{n}=(ab)^{n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8bc1ff793158ec97e1cd021942246bd7c5ff7d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:13.405ex; height:2.843ex;" alt="{\displaystyle a^{n}b^{n}=(ab)^{n}\,\!}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\tfrac {a}{b}})^{n}={\tfrac {a^{n}}{b^{n}}}\qquad (b\neq 0)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>b</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\tfrac {a}{b}})^{n}={\tfrac {a^{n}}{b^{n}}}\qquad (b\neq 0)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/112bedabca5dd1af39f68fa49aceb5f3074a3fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-right: -0.387ex; width:22.603ex; height:3.843ex;" alt="{\displaystyle ({\tfrac {a}{b}})^{n}={\tfrac {a^{n}}{b^{n}}}\qquad (b\neq 0)\,\!}"></span></li></ol> <div class="mw-heading mw-heading3"><h3 id="Eksponenttina_nolla">Eksponenttina nolla</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=5" title="Muokkaa osiota Eksponenttina nolla" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=5" title="Muokkaa osion lähdekoodia: Eksponenttina nolla"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Potenssin tulkinta kertolaskun kautta ei kerro, mitä luvun nollas potenssi olisi: eihän ole olemassa tuloa, jossa on 0 tulon tekijää. Mikäli halutaan, että luku voidaan korottaa myös nollanteen potenssiin, täytyy sopia, mitä nollannella potenssilla tarkoitetaan. </p><p>Periaatteessa tämä sopimus voitaisiin tehdä täysin mielivaltaisesti, mutta useimmissa tapauksissa edellä esitetyt potenssin laskusäännöt eivät pätisi nollansilla potensseilla. Kun sovelletaan toista laskusääntöä potenssiin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a894726dd8a070c22ea9b3b52e93b3840b7c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.676ex;" alt="{\displaystyle a^{0}}"></span>, jossa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> on nollasta eroava reaaliluku, saadaan </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{0}=a^{1-1}={\tfrac {a^{1}}{a^{1}}}={\tfrac {a}{a}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>a</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{0}=a^{1-1}={\tfrac {a^{1}}{a^{1}}}={\tfrac {a}{a}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/289c23aa898ff9377914e34f0400014ea4312c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:24.468ex; height:4.343ex;" alt="{\displaystyle a^{0}=a^{1-1}={\tfrac {a^{1}}{a^{1}}}={\tfrac {a}{a}}=1}"></span>. </p> </center> <p>Siis luvun nollannen potenssin on oltava aina 1, mikäli halutaan laskusäännön <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c52fee07c5d0052fb988540513e4ba1af65ea72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.309ex; height:3.676ex;" alt="{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}}"></span> pätevän myös tapauksessa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c9d8e54796e7de7d4738510cc10bc3fc55d48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle m=n}"></span>. Siksi määritellään </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/448ca9a3f4ef03c4dfcf69258912d2c90b097842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.545ex; height:2.676ex;" alt="{\displaystyle a^{0}=1}"></span> </p> </center> <p>kaikilla nollasta eroavilla reaaliluvuilla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. Näin määritellen myös muut potenssin laskusäännöt ovat voimassa nollansille potensseille. </p><p>Luvun nolla nollannelle potenssille laskusäännöt eivät kuitenkaan anna vastaavia rajoitteita. Siksi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106f0c4e1cbccbfcbb61001a8c17b8427c65366d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 0^{0}}"></span> onkin <a href="/wiki/Ep%C3%A4m%C3%A4%C3%A4r%C3%A4inen_muoto" title="Epämääräinen muoto">epämääräinen muoto</a> eli se jätetään yleisesti määrittelemättä. Joissain erikoistapauksissa kuten <a href="/wiki/Binomikaava" class="mw-redirect" title="Binomikaava">binomikaavan</a> ja <a href="/wiki/Potenssisarja" title="Potenssisarja">potenssisarjojen</a> yhteydessä määritellään kuitenkin toisinaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478100bc5766c6af537439ef9309f9ddf2f9a6ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 0^{0}=1}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Negatiivinen_eksponentti">Negatiivinen eksponentti</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=6" title="Muokkaa osiota Negatiivinen eksponentti" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=6" title="Muokkaa osion lähdekoodia: Negatiivinen eksponentti"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Samoin kuin nollas potenssi määritellään myös negatiiviset kokonaislukupotenssit pyrkimällä säilyttämään potenssin laskusäännöt. Olkoon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> positiivinen kokonaisluku ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> nollasta eroava. Jotta sääntö <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c52fee07c5d0052fb988540513e4ba1af65ea72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.309ex; height:3.676ex;" alt="{\displaystyle {\tfrac {a^{m}}{a^{n}}}=a^{m-n}}"></span> pätisi myös, kun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490c01b0cb770144f28afd17bb5fef277daf6f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle m<n}"></span>, tulee olla </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-n}=a^{0-n}={\tfrac {a^{0}}{a^{n}}}={\tfrac {1}{a^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-n}=a^{0-n}={\tfrac {a^{0}}{a^{n}}}={\tfrac {1}{a^{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b55e08b12a018ac151a315ec97898687a603c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.559ex; height:4.009ex;" alt="{\displaystyle a^{-n}=a^{0-n}={\tfrac {a^{0}}{a^{n}}}={\tfrac {1}{a^{n}}}.}"></span> </p> </center> <p>Toisin sanoen määritellään luvun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>:s negatiivinen kokonaislukupotenssi luvun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2687184a7698e75db65a25bea7afd207bff3d03b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.448ex; height:2.343ex;" alt="{\displaystyle a^{n}}"></span> käänteisluvuksi. Näin määritellen ovat muutkin potenssin laskusäännöt voimassa negatiivisen kokonaislukueksponentin tapauksessa. </p> <div class="mw-heading mw-heading3"><h3 id="Eksponenttina_rationaaliluku">Eksponenttina rationaaliluku</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=7" title="Muokkaa osiota Eksponenttina rationaaliluku" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=7" title="Muokkaa osion lähdekoodia: Eksponenttina rationaaliluku"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seuraavaksi yleistetään potenssin käsite kaikille <a href="/wiki/Rationaaliluku" title="Rationaaliluku">rationaalisille</a> eksponenteille, jotta voidaan puhua esimerkiksi potensseista <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33b3e1d440232e8b08923ef872d21f04e9146763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.898ex; height:3.509ex;" alt="{\displaystyle 2^{\frac {1}{3}}}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{-{\frac {5}{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{-{\frac {5}{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5e1d8c2adcbd7dedeaf9000d6a0308ff427d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.177ex; height:3.509ex;" alt="{\displaystyle 3^{-{\frac {5}{3}}}}"></span>. Vaaditaan yhä, että edellä esitellyt potenssin laskusäännöt säilyvät voimassa. </p><p>Olkoon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> positiivinen kokonaisluku ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> positiivinen reaaliluku. Laskusäännön <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{m})^{n}=a^{mn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{m})^{n}=a^{mn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b430c76fcf7df71be0da8b4e721f4189e41c7ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.922ex; height:2.843ex;" alt="{\displaystyle (a^{m})^{n}=a^{mn}}"></span> nojalla on määriteltävä siten, että </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{\frac {1}{n}})^{n}=a^{\frac {n}{n}}=a^{1}=a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{\frac {1}{n}})^{n}=a^{\frac {n}{n}}=a^{1}=a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5228dc1f3b916e061d09679730f50ea2e051a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.682ex; height:3.843ex;" alt="{\displaystyle (a^{\frac {1}{n}})^{n}=a^{\frac {n}{n}}=a^{1}=a.}"></span> </p> </center> <p>Siis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\frac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d741d942d62af470be1751f21021e8a140f845a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.099ex; height:3.343ex;" alt="{\displaystyle a^{\frac {1}{n}}}"></span> on se luku, jonka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>:s potenssi on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> itse. Tällaista lukua kutsutaan luvun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>:nneksi <a href="/wiki/Juuri_(laskutoimitus)" title="Juuri (laskutoimitus)">juureksi</a>. Määritellään sen tähden </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\frac {1}{n}}={\sqrt[{n}]{a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {1}{n}}={\sqrt[{n}]{a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/051c662b13008285b4f06ff9b561017b815ac20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.01ex; height:4.009ex;" alt="{\displaystyle a^{\frac {1}{n}}={\sqrt[{n}]{a}}.}"></span> </p> </center> <p>Olkoon sitten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> mikä tahansa kokonaisluku. Vaatimalla, että potenssin potenssia koskeva laskusääntö pätee myös potenssille <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\frac {m}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {m}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a712a805a8e616dd50760a69c159450bf16e59c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.47ex; height:3.009ex;" alt="{\displaystyle a^{\frac {m}{n}}}"></span>, saadaan </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\frac {m}{n}}=(a^{m})^{\frac {1}{n}}={\sqrt[{n}]{a^{m}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {m}{n}}=(a^{m})^{\frac {1}{n}}={\sqrt[{n}]{a^{m}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6881843dcf9803a7132f3c6f4592297082cd7e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.737ex; height:3.843ex;" alt="{\displaystyle a^{\frac {m}{n}}=(a^{m})^{\frac {1}{n}}={\sqrt[{n}]{a^{m}}}.}"></span> </p> </center> <p>Tämän mukaisesti määritellään siis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{\frac {m}{n}}={\sqrt[{n}]{a^{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {m}{n}}={\sqrt[{n}]{a^{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb409de8f30f86d2ee48c97de1f939fdd257d916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.409ex; height:3.343ex;" alt="{\displaystyle a^{\frac {m}{n}}={\sqrt[{n}]{a^{m}}}}"></span> kaikilla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0,m\in \mathbb {Z} ,n\in \mathbb {Z} _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0,m\in \mathbb {Z} ,n\in \mathbb {Z} _{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8585dfdaa8ab20e2e2088dedf2518d5778c00954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.286ex; height:2.509ex;" alt="{\displaystyle a>0,m\in \mathbb {Z} ,n\in \mathbb {Z} _{+}}"></span>. Myös kaikki muut potenssin laskusäännöt ovat voimassa tällaisella rationaalisen eksponentin määrittelyllä. </p> <div class="mw-heading mw-heading3"><h3 id="Miksi_kantaluvun_on_oltava_positiivinen?"><span id="Miksi_kantaluvun_on_oltava_positiivinen.3F"></span>Miksi kantaluvun on oltava positiivinen?</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=8" title="Muokkaa osiota Miksi kantaluvun on oltava positiivinen?" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=8" title="Muokkaa osion lähdekoodia: Miksi kantaluvun on oltava positiivinen?"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rationaalisen eksponentin tapauksessa on esitetty rajoitus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}"></span>. Siis esimerkiksi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{1/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{1/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b788642496d5b1c9d7fc4da149e25726f3d64ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.478ex; height:3.343ex;" alt="{\displaystyle (-1)^{1/3}}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{-1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/655cd136931697608892733758ca81224310ec5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.139ex; height:2.843ex;" alt="{\displaystyle 0^{-1/2}}"></span> eivät ole määriteltyjä lausekkeita. Jos kantaluvulle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> sallittaisiin negatiivisia arvoja, jouduttaisiin seuraavanlaiseen ristiriitaan: </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1={\sqrt[{3}]{-1}}=(-1)^{\frac {1}{3}}=(-1)^{\frac {2}{6}}={\sqrt[{6}]{(-1)^{2}}}={\sqrt[{6}]{1}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>6</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1={\sqrt[{3}]{-1}}=(-1)^{\frac {1}{3}}=(-1)^{\frac {2}{6}}={\sqrt[{6}]{(-1)^{2}}}={\sqrt[{6}]{1}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8577a21e50817b68b887a6eb979f59cf67baad4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:52.565ex; height:4.843ex;" alt="{\displaystyle -1={\sqrt[{3}]{-1}}=(-1)^{\frac {1}{3}}=(-1)^{\frac {2}{6}}={\sqrt[{6}]{(-1)^{2}}}={\sqrt[{6}]{1}}=1.}"></span> </p> </center> <p>Koska <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\neq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6d4d5cd6ce38b695c4b313724e0b3926d1ae12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.232ex; height:2.676ex;" alt="{\displaystyle -1\neq 1}"></span>, on joko kiellettävä murtolukueksponenttien laventaminen (ja myös supistaminen) tai sitten rajoituttava vain ei-negatiivisiin kantalukuihin. Jälkimmäinen valinta on luonnollisempi. </p><p>Myöskään nolla ei ole sovelias arvo rationaalipotenssin kantaluvulle. Jos nimittäin eksponentti on negatiivinen, päädytään <a href="/wiki/Nollalla_jakaminen" title="Nollalla jakaminen">jakamaan nollalla</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Eksponenttina_irrationaalinen_luku">Eksponenttina irrationaalinen luku</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=9" title="Muokkaa osiota Eksponenttina irrationaalinen luku" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=9" title="Muokkaa osion lähdekoodia: Eksponenttina irrationaalinen luku"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Potenssiinkorotus on edellä määritelty siten, että eksponentti voi olla mikä <a href="/wiki/Rationaaliluku" title="Rationaaliluku">rationaaliluku</a> hyvänsä. Voidaan osoittaa, että mitä tahansa <a href="/wiki/Irrationaaliluku" title="Irrationaaliluku">irrationaalilukua</a> voidaan arvioida mielivaltaisen tarkasti rationaaliluvuilla. Siksi jokaista irrationaalilukua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> kohden on olemassa rationaalilukujen jono <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1},q_{2},q_{3},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1},q_{2},q_{3},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d845a6511712f448df3511de4dd7e9f0a28f1aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.099ex; height:2.009ex;" alt="{\displaystyle q_{1},q_{2},q_{3},\dots }"></span> siten, että jono suppenee kohti lukua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. Tällöin myös jono <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{q_{1}},a^{q_{2}},a^{q_{3}},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{q_{1}},a^{q_{2}},a^{q_{3}},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db00b49a0a13c3c79caee71b571d8291014f815" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.906ex; height:2.676ex;" alt="{\displaystyle a^{q_{1}},a^{q_{2}},a^{q_{3}},\dots }"></span> suppenee riippumatta positiivisesta reaaliluvusta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. Irrationaalinen potenssi voidaan täten määritellä raja-arvona </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{r}=\lim _{i\rightarrow \infty }a^{q_{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{r}=\lim _{i\rightarrow \infty }a^{q_{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b9cad42a211ffb305684f6c540af2c3b5895d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.01ex; height:4.009ex;" alt="{\displaystyle a^{r}=\lim _{i\rightarrow \infty }a^{q_{i}}.}"></span> </p> </center> <p>Voidaan osoittaa, että potenssin laskusäännöt ovat voimassa myös irrationaalisen eksponentin tapauksessa. Näin on potenssiinkorotus määritelty kaikilla eksponentin reaalisilla arvoilla. Irrationaalinen eksponentti voidaan määritellä myös <a href="/wiki/Infimum" title="Infimum">infimumin</a> ja <a href="/wiki/Supremum" title="Supremum">supremumin</a> avulla seuraavasti. Olkoon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> irrationaaliluku. Kun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5b9d9fb0ff9d4455e75ccd29676bd7f33da80e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>1}"></span>, määritellään </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{r}=\inf\{a^{q}\,|\,q\in \mathbb {Q} \ {\textrm {ja}}\ q>r\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ja</mtext> </mrow> </mrow> <mtext> </mtext> <mi>q</mi> <mo>></mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{r}=\inf\{a^{q}\,|\,q\in \mathbb {Q} \ {\textrm {ja}}\ q>r\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c464cc07a758aad6d07631f17cf970e5549c120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.535ex; height:2.843ex;" alt="{\displaystyle a^{r}=\inf\{a^{q}\,|\,q\in \mathbb {Q} \ {\textrm {ja}}\ q>r\}.}"></span> </p> </center> <p>Kun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<a<1\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a<1\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f08a6cf0e6f80370dda05848af2d766502f9cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:10.139ex; height:2.176ex;" alt="{\displaystyle 0<a<1\,\!}"></span>, määritellään </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{r}=\sup\{a^{q}\,|\,q\in \mathbb {Q} \ {\textrm {ja}}\ q>r\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ja</mtext> </mrow> </mrow> <mtext> </mtext> <mi>q</mi> <mo>></mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{r}=\sup\{a^{q}\,|\,q\in \mathbb {Q} \ {\textrm {ja}}\ q>r\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c40c9be1ec27ba2fe57f42a05bedad4c54435c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.385ex; height:2.843ex;" alt="{\displaystyle a^{r}=\sup\{a^{q}\,|\,q\in \mathbb {Q} \ {\textrm {ja}}\ q>r\}.}"></span> </p> </center> <div class="mw-heading mw-heading2"><h2 id="Potenssiin_perustuvia_funktioita">Potenssiin perustuvia funktioita</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=10" title="Muokkaa osiota Potenssiin perustuvia funktioita" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=10" title="Muokkaa osion lähdekoodia: Potenssiin perustuvia funktioita"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Potenssifunktio" title="Potenssifunktio">Potenssifunktiossa</a> potenssimerkinnän kantaluku on muuttuja ja eksponentti vakio. Potenssifunktiot ovat yksinkertaisia funktioita, joilla on kuitenkin lukuisia sovelluksia mallinnuksessa. <a href="/wiki/Eksponenttifunktio" title="Eksponenttifunktio">Eksponenttifunktiossa</a> potenssimerkinnän eksponentti on muuttuja ja kantaluku vakio. Myös eksponenttifunktiolla on monia sovelluksia, minkä takia näitä funktioita voidaan pitää tärkeimpinä yleisfunktioina matematiikassa. </p><p><a href="/wiki/Fermat%E2%80%99n_pieni_lause" title="Fermat’n pieni lause">Fermat'n pienen lauseen</a> perusteella kaikilla kokonaisluvun potenssiluvuilla on myös se ominaisuus, että vähentämällä jonkin kokonaisluvun potenssista yksi saadaan <a href="/wiki/Yhdistetty_luku" title="Yhdistetty luku">yhdistetty luku</a>, joka on jaollinen potenssin juurta yhtä pienemmällä luvulla. Esimerkiksi kaikista 18:n potensseista saadaan vähentämällä yksi jokin 17:llä jaollinen luku, esimerkiksi:<sup><i><a href="/wiki/Wikipedia:Merkitse_l%C3%A4hteet" title="Wikipedia:Merkitse lähteet">lähde?</a></i></sup> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5832=18^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5832</mn> <mo>=</mo> <msup> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5832=18^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b016a868a7fd61a17b9509b5cb009b406c10da40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.127ex; height:2.676ex;" alt="{\displaystyle 5832=18^{3}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5832-1=5831}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5832</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>5831</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5832-1=5831}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/876674cff5a3815b66365f0b2a2a82ac70da2645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.401ex; height:2.343ex;" alt="{\displaystyle 5832-1=5831}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5831=17\cdot 343}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5831</mn> <mo>=</mo> <mn>17</mn> <mo>⋅</mo> <mn>343</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5831=17\cdot 343}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c268f2d6dd5bf410829e5c109aa89192b2bd998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.24ex; height:2.176ex;" alt="{\displaystyle 5831=17\cdot 343}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Katso_myös"><span id="Katso_my.C3.B6s"></span>Katso myös</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=11" title="Muokkaa osiota Katso myös" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=11" title="Muokkaa osion lähdekoodia: Katso myös"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kahden_potenssit" title="Kahden potenssit">Kahden potenssit</a></li> <li><a href="/wiki/Imaginaariyksikk%C3%B6" title="Imaginaariyksikkö">Imaginaariyksikkö</a></li> <li><a href="/wiki/Tieteellinen_merkint%C3%A4tapa" class="mw-redirect" title="Tieteellinen merkintätapa">Tieteellinen merkintätapa</a></li> <li><a href="/wiki/Tetraatio" title="Tetraatio">Tetraatio</a></li> <li><a href="/wiki/Knuthin_nuolinotaatio" title="Knuthin nuolinotaatio">Knuthin nuolinotaatio</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lähteet"><span id="L.C3.A4hteet"></span>Lähteet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Potenssi&veaction=edit&section=12" title="Muokkaa osiota Lähteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Potenssi&action=edit&section=12" title="Muokkaa osion lähdekoodia: Lähteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="viitteet-malline" class="viitteet-malline" style="list-style-type:decimal;"><ol class="references"> <li id="cite_note-r1-1"><span class="mw-cite-backlink"><a href="#cite_ref-r1_1-0">↑</a></span> <span class="reference-text"><span class="kirjaviite" title="Kirjaviite">Rikkonen, Harri: <i>Matematiikan pitkä peruskurssi II: Reaalimuuttujan funktioiden differentiaalilasku</i>.  Helsinki:  Otakustantamo, 1969.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/951-671-022-0" title="Toiminnot:Kirjalähteet/951-671-022-0">ISBN 951-671-022-0</a> </span></span> </li> </ol> </div> <p><i style="display:none; speak:none;"> </i> </p></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Noudettu kohteesta ”<a dir="ltr" href="https://fi.wikipedia.org/w/index.php?title=Potenssi&oldid=22804300">https://fi.wikipedia.org/w/index.php?title=Potenssi&oldid=22804300</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Toiminnot:Luokat" title="Toiminnot:Luokat">Luokka</a>: <ul><li><a href="/wiki/Luokka:Alkeisalgebra" title="Luokka:Alkeisalgebra">Alkeisalgebra</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Piilotetut luokat: <ul><li><a href="/wiki/Luokka:Puutteelliset_l%C3%A4hdemerkinn%C3%A4t" title="Luokka:Puutteelliset lähdemerkinnät">Puutteelliset lähdemerkinnät</a></li><li><a href="/wiki/Luokka:Seulonnan_keskeiset_artikkelit" title="Luokka:Seulonnan keskeiset artikkelit">Seulonnan keskeiset artikkelit</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigointivalikko</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Henkilökohtaiset työkalut</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="IP-osoitteesi käyttäjäsivu">Et ole kirjautunut</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Toiminnot:Oma_keskustelu" title="Keskustelu tämän IP-osoitteen muokkauksista [n]" accesskey="n"><span>Keskustelu</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Toiminnot:Omat_muokkaukset" title="Luettelo tästä IP-osoitteesta tehdyistä muokkauksista [y]" accesskey="y"><span>Muokkaukset</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:Luo_tunnus&returnto=Potenssi" title="On suositeltavaa luoda käyttäjätunnus ja kirjautua sisään. 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<nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Työkalut</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Toiminnot:T%C3%A4nne_viittaavat_sivut/Potenssi" title="Lista sivuista, jotka viittaavat tänne [j]" accesskey="j"><span>Tänne viittaavat sivut</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Toiminnot:Linkitetyt_muutokset/Potenssi" rel="nofollow" title="Viimeisimmät muokkaukset sivuissa, joille viitataan tältä sivulta [k]" accesskey="k"><span>Linkitettyjen sivujen muutokset</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Toiminnot:Toimintosivut" title="Näytä toimintosivut [q]" accesskey="q"><span>Toimintosivut</span></a></li><li id="t-permalink" class="mw-list-item"><a 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<ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Exponentiation" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikifunctions mw-list-item"><a href="https://www.wikifunctions.org/wiki/Z12665" hreflang="en"><span>Wikifunctions</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q33456" title="Linkki yhdistettyyn keskustietovaraston kohteeseen [g]" accesskey="g"><span>Wikidata-kohde</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Muilla kielillä</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Magsverheffing" title="Magsverheffing — afrikaans" lang="af" hreflang="af" data-title="Magsverheffing" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Potenz_(Mathematik)" title="Potenz (Mathematik) — sveitsinsaksa" lang="gsw" hreflang="gsw" data-title="Potenz (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="sveitsinsaksa" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%95%E1%88%B4%E1%89%B5" title="ንሴት — amhara" lang="am" hreflang="am" data-title="ንሴት" data-language-autonym="አማርኛ" data-language-local-name="amhara" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%81%D8%B9_%D8%A3%D8%B3%D9%8A" title="رفع أسي — arabia" lang="ar" hreflang="ar" data-title="رفع أسي" data-language-autonym="العربية" data-language-local-name="arabia" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Potenciaci%C3%B3n" title="Potenciación — asturia" lang="ast" hreflang="ast" data-title="Potenciación" data-language-autonym="Asturianu" data-language-local-name="asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Q%C3%BCvv%C9%99t%C9%99_y%C3%BCks%C9%99ltm%C9%99" title="Qüvvətə yüksəltmə — azeri" lang="az" hreflang="az" data-title="Qüvvətə yüksəltmə" data-language-autonym="Azərbaycanca" data-language-local-name="azeri" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Eksponensiasi" title="Eksponensiasi — indonesia" lang="id" hreflang="id" data-title="Eksponensiasi" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pengeksponenan" title="Pengeksponenan — malaiji" lang="ms" hreflang="ms" data-title="Pengeksponenan" data-language-autonym="Bahasa Melayu" data-language-local-name="malaiji" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%82%E0%A6%9A%E0%A6%95%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="সূচকীকরণ — bengali" lang="bn" hreflang="bn" data-title="সূচকীকরণ" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%94%D3%99%D1%80%D3%99%D0%B6%D3%99%D0%B3%D3%99_%D0%BA%D2%AF%D1%82%D3%99%D1%80%D0%B5%D2%AF" title="Дәрәжәгә күтәреү — baškiiri" lang="ba" hreflang="ba" data-title="Дәрәжәгә күтәреү" data-language-autonym="Башҡортса" data-language-local-name="baškiiri" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%82%D1%83%D0%BF%D0%B5%D0%BD%D1%8F%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Ступеняванне — valkovenäjä" lang="be" hreflang="be" data-title="Ступеняванне" data-language-autonym="Беларуская" data-language-local-name="valkovenäjä" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Eksponentasyon" title="Eksponentasyon — Central Bikol" lang="bcl" hreflang="bcl" data-title="Eksponentasyon" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Eksponent" title="Eksponent — bosnia" lang="bs" hreflang="bs" data-title="Eksponent" data-language-autonym="Bosanski" data-language-local-name="bosnia" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%83%D0%B2%D0%B0%D0%BD%D0%B5_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Степенуване (математика) — bulgaria" lang="bg" hreflang="bg" data-title="Степенуване (математика)" data-language-autonym="Български" data-language-local-name="bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%97%D1%8D%D1%80%D0%B3%D1%8D%D0%B4%D1%8D_%D0%B4%D1%8D%D0%B1%D0%B6%D2%AF%D2%AF%D0%BB%D1%85%D1%8D" title="Зэргэдэ дэбжүүлхэ — Russia Buriat" lang="bxr" hreflang="bxr" data-title="Зэргэдэ дэбжүүлхэ" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Potenciaci%C3%B3" title="Potenciació — katalaani" lang="ca" hreflang="ca" data-title="Potenciació" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BF%D0%B0%D1%88%D1%82%D0%B0%D1%80%D1%83" title="Капаштару — tšuvassi" lang="cv" hreflang="cv" data-title="Капаштару" data-language-autonym="Чӑвашла" data-language-local-name="tšuvassi" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Umoc%C5%88ov%C3%A1n%C3%AD" title="Umocňování — tšekki" lang="cs" hreflang="cs" data-title="Umocňování" data-language-autonym="Čeština" data-language-local-name="tšekki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Kutambanura_(nhamba)" title="Kutambanura (nhamba) — šona" lang="sn" hreflang="sn" data-title="Kutambanura (nhamba)" data-language-autonym="ChiShona" data-language-local-name="šona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Esbonydd" title="Esbonydd — kymri" lang="cy" hreflang="cy" data-title="Esbonydd" data-language-autonym="Cymraeg" data-language-local-name="kymri" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Potens_(matematik)" title="Potens (matematik) — tanska" lang="da" hreflang="da" data-title="Potens (matematik)" data-language-autonym="Dansk" data-language-local-name="tanska" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Potenz_(Mathematik)" title="Potenz (Mathematik) — saksa" lang="de" hreflang="de" data-title="Potenz (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="saksa" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Astendamine" title="Astendamine — viro" lang="et" hreflang="et" data-title="Astendamine" data-language-autonym="Eesti" data-language-local-name="viro" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%8D%CE%BD%CE%B1%CE%BC%CE%B7_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Δύναμη (μαθηματικά) — kreikka" lang="el" hreflang="el" data-title="Δύναμη (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="kreikka" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Exponentiation" title="Exponentiation — englanti" lang="en" hreflang="en" data-title="Exponentiation" data-language-autonym="English" data-language-local-name="englanti" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Potenciaci%C3%B3n" title="Potenciación — espanja" lang="es" hreflang="es" data-title="Potenciación" data-language-autonym="Español" data-language-local-name="espanja" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Potenco_(matematiko)" title="Potenco (matematiko) — esperanto" lang="eo" hreflang="eo" data-title="Potenco (matematiko)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Berreketa" title="Berreketa — baski" lang="eu" hreflang="eu" data-title="Berreketa" data-language-autonym="Euskara" data-language-local-name="baski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%A7%D9%86_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="توان (ریاضی) — persia" lang="fa" hreflang="fa" data-title="توان (ریاضی)" data-language-autonym="فارسی" data-language-local-name="persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Potensur" title="Potensur — fääri" lang="fo" hreflang="fo" data-title="Potensur" data-language-autonym="Føroyskt" data-language-local-name="fääri" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Exponentiation" title="Exponentiation — ranska" lang="fr" hreflang="fr" data-title="Exponentiation" data-language-autonym="Français" data-language-local-name="ranska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Easp%C3%B3nant" title="Easpónant — iiri" lang="ga" hreflang="ga" data-title="Easpónant" data-language-autonym="Gaeilge" data-language-local-name="iiri" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Potenciaci%C3%B3n" title="Potenciación — galicia" lang="gl" hreflang="gl" data-title="Potenciación" data-language-autonym="Galego" data-language-local-name="galicia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%86%AA" title="冪 — gan-kiina" lang="gan" hreflang="gan" data-title="冪" data-language-autonym="贛語" data-language-local-name="gan-kiina" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%98%D0%B4%D1%80%D0%B8%D0%BB%D2%BB%D0%B0%D0%BD" title="Идрилһан — kalmukki" lang="xal" hreflang="xal" data-title="Идрилһан" data-language-autonym="Хальмг" data-language-local-name="kalmukki" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B1%B0%EB%93%AD%EC%A0%9C%EA%B3%B1" title="거듭제곱 — korea" lang="ko" hreflang="ko" data-title="거듭제곱" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%BD%D5%BF%D5%AB%D5%B3%D5%A1%D5%B6_(%D5%B0%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%AB%D5%BE)" title="Աստիճան (հանրահաշիվ) — armenia" lang="hy" hreflang="hy" data-title="Աստիճան (հանրահաշիվ)" data-language-autonym="Հայերեն" data-language-local-name="armenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%98%E0%A4%BE%E0%A4%A4%E0%A4%BE%E0%A4%82%E0%A4%95" title="घातांक — hindi" lang="hi" hreflang="hi" data-title="घातांक" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Potenciranje" title="Potenciranje — kroatia" lang="hr" hreflang="hr" data-title="Potenciranje" data-language-autonym="Hrvatski" data-language-local-name="kroatia" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Potenco" title="Potenco — ido" lang="io" hreflang="io" data-title="Potenco" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Potentiation" title="Potentiation — interlingua" lang="ia" hreflang="ia" data-title="Potentiation" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Veldi_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Veldi (stærðfræði) — islanti" lang="is" hreflang="is" data-title="Veldi (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="islanti" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Potenza_(matematica)" title="Potenza (matematica) — italia" lang="it" hreflang="it" data-title="Potenza (matematica)" data-language-autonym="Italiano" data-language-local-name="italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he badge-Q17437796 badge-featuredarticle mw-list-item" title="suositeltu artikkeli"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%96%D7%A7%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="חזקה (מתמטיקה) — heprea" lang="he" hreflang="he" data-title="חזקה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="heprea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D3%99%D1%80%D0%B5%D0%B6%D0%B5%D0%BB%D0%B5%D1%83" title="Дәрежелеу — kazakki" lang="kk" hreflang="kk" data-title="Дәрежелеу" data-language-autonym="Қазақша" data-language-local-name="kazakki" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Eksponansyasyon" title="Eksponansyasyon — Guianan Creole" lang="gcr" hreflang="gcr" data-title="Eksponansyasyon" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Potentia_(mathematica)" title="Potentia (mathematica) — latina" lang="la" hreflang="la" data-title="Potentia (mathematica)" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/K%C4%81pin%C4%81%C5%A1ana" title="Kāpināšana — latvia" lang="lv" hreflang="lv" data-title="Kāpināšana" data-language-autonym="Latviešu" data-language-local-name="latvia" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/K%C4%97limas_laipsniu" title="Kėlimas laipsniu — liettua" lang="lt" hreflang="lt" data-title="Kėlimas laipsniu" data-language-autonym="Lietuvių" data-language-local-name="liettua" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Machsverh%C3%B6ffing" title="Machsverhöffing — limburg" lang="li" hreflang="li" data-title="Machsverhöffing" data-language-autonym="Limburgs" data-language-local-name="limburg" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Esponenti" title="Esponenti — lingua franca nova" lang="lfn" hreflang="lfn" data-title="Esponenti" data-language-autonym="Lingua Franca Nova" data-language-local-name="lingua franca nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hatv%C3%A1ny" title="Hatvány — unkari" lang="hu" hreflang="hu" data-title="Hatvány" data-language-autonym="Magyar" data-language-local-name="unkari" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Степенување — makedonia" lang="mk" hreflang="mk" data-title="Степенување" data-language-autonym="Македонски" data-language-local-name="makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Toraka_(matematika)" title="Toraka (matematika) — malagassi" lang="mg" hreflang="mg" data-title="Toraka (matematika)" data-language-autonym="Malagasy" data-language-local-name="malagassi" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Machtsverheffen" title="Machtsverheffen — hollanti" lang="nl" hreflang="nl" data-title="Machtsverheffen" data-language-autonym="Nederlands" data-language-local-name="hollanti" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%98%E0%A4%BE%E0%A4%A4%E0%A4%BE%E0%A4%99%E0%A5%8D%E0%A4%95" title="घाताङ्क — nepali" lang="ne" hreflang="ne" data-title="घाताङ्क" data-language-autonym="नेपाली" data-language-local-name="nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%AA%E4%B9%97" title="冪乗 — japani" lang="ja" hreflang="ja" data-title="冪乗" data-language-autonym="日本語" data-language-local-name="japani" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Potens" title="Potens — pohjoisfriisi" lang="frr" hreflang="frr" data-title="Potens" data-language-autonym="Nordfriisk" data-language-local-name="pohjoisfriisi" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Potens_(matematikk)" title="Potens (matematikk) — norjan bokmål" lang="nb" hreflang="nb" data-title="Potens (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norjan bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Potens_i_matematikk" title="Potens i matematikk — norjan nynorsk" lang="nn" hreflang="nn" data-title="Potens i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norjan nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Aangessoo(ekispoonentii)" title="Aangessoo(ekispoonentii) — oromo" lang="om" hreflang="om" data-title="Aangessoo(ekispoonentii)" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%98%E0%A8%BE%E0%A8%A4_%E0%A8%85%E0%A9%B0%E0%A8%95" title="ਘਾਤ ਅੰਕ — pandžabi" lang="pa" hreflang="pa" data-title="ਘਾਤ ਅੰਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Exponenshieshan" title="Exponenshieshan — jamaikankreolienglanti" lang="jam" hreflang="jam" data-title="Exponenshieshan" data-language-autonym="Patois" data-language-local-name="jamaikankreolienglanti" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pot%C4%99gowanie" title="Potęgowanie — puola" lang="pl" hreflang="pl" data-title="Potęgowanie" data-language-autonym="Polski" data-language-local-name="puola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Exponencia%C3%A7%C3%A3o" title="Exponenciação — portugali" lang="pt" hreflang="pt" data-title="Exponenciação" data-language-autonym="Português" data-language-local-name="portugali" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Putere_(matematic%C4%83)" title="Putere (matematică) — romania" lang="ro" hreflang="ro" data-title="Putere (matematică)" data-language-autonym="Română" data-language-local-name="romania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Yupa_huqariy" title="Yupa huqariy — ketšua" lang="qu" hreflang="qu" data-title="Yupa huqariy" data-language-autonym="Runa Simi" data-language-local-name="ketšua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%BE%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B2_%D1%81%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%8C" title="Возведение в степень — venäjä" lang="ru" hreflang="ru" data-title="Возведение в степень" data-language-autonym="Русский" data-language-local-name="venäjä" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%91%D2%AF%D1%82%D2%AF%D0%BD_%D0%BA%D3%A9%D1%80%D0%B4%D3%A9%D1%80%D3%A9%D3%A9%D1%87%D1%87%D2%AF%D0%BB%D1%8D%D1%8D%D1%85_%D1%81%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%8C" title="Бүтүн көрдөрөөччүлээх степень — jakuutti" lang="sah" hreflang="sah" data-title="Бүтүн көрдөрөөччүлээх степень" data-language-autonym="Саха тыла" data-language-local-name="jakuutti" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Putenza_(matim%C3%A0tica)" title="Putenza (matimàtica) — sisilia" lang="scn" hreflang="scn" data-title="Putenza (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="sisilia" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Exponentiation" title="Exponentiation — Simple English" lang="en-simple" hreflang="en-simple" data-title="Exponentiation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Umoc%C5%88ovanie" title="Umocňovanie — slovakki" lang="sk" hreflang="sk" data-title="Umocňovanie" data-language-autonym="Slovenčina" data-language-local-name="slovakki" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Potenciranje" title="Potenciranje — sloveeni" lang="sl" hreflang="sl" data-title="Potenciranje" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%D9%88%D8%A7%D9%86_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="توان (ماتماتیک) — soranî" lang="ckb" hreflang="ckb" data-title="توان (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="soranî" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D0%BE%D0%B2%D0%B0%D1%9A%D0%B5" title="Степеновање — serbia" lang="sr" hreflang="sr" data-title="Степеновање" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Stepenovanje" title="Stepenovanje — serbokroaatti" lang="sh" hreflang="sh" data-title="Stepenovanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaatti" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Potens" title="Potens — ruotsi" lang="sv" hreflang="sv" data-title="Potens" data-language-autonym="Svenska" data-language-local-name="ruotsi" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pagpapalakas_(matematika)" title="Pagpapalakas (matematika) — tagalog" lang="tl" hreflang="tl" data-title="Pagpapalakas (matematika)" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="அடுக்கேற்றம் — tamili" lang="ta" hreflang="ta" data-title="அடுக்கேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="tamili" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th badge-Q17437798 badge-goodarticle mw-list-item" title="hyvä artikkeli"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A2%E0%B8%81%E0%B8%81%E0%B8%B3%E0%B8%A5%E0%B8%B1%E0%B8%87" title="การยกกำลัง — thai" lang="th" hreflang="th" data-title="การยกกำลัง" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C5%A9y_th%E1%BB%ABa" title="Lũy thừa — vietnam" lang="vi" hreflang="vi" data-title="Lũy thừa" data-language-autonym="Tiếng Việt" data-language-local-name="vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9Cs" title="Üs — turkki" lang="tr" hreflang="tr" data-title="Üs" data-language-autonym="Türkçe" data-language-local-name="turkki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B4%D0%BD%D0%B5%D1%81%D0%B5%D0%BD%D0%BD%D1%8F_%D0%B4%D0%BE_%D1%81%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%8F" title="Піднесення до степеня — ukraina" lang="uk" hreflang="uk" data-title="Піднесення до степеня" data-language-autonym="Українська" data-language-local-name="ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D8%AF%DB%95%D8%B1%D9%89%D8%AC%DB%95_(%D9%85%D8%A7%D8%AA%DB%90%D9%85%D8%A7%D8%AA%D9%89%D9%83%D8%A7)" title="دەرىجە (ماتېماتىكا) — uiguuri" lang="ug" hreflang="ug" data-title="دەرىجە (ماتېماتىكا)" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="uiguuri" class="interlanguage-link-target"><span>ئۇيغۇرچە / Uyghurche</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Eksponentasyon" title="Eksponentasyon — waray" lang="war" hreflang="war" data-title="Eksponentasyon" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%B9%82" title="幂 — wu-kiina" lang="wuu" hreflang="wuu" data-title="幂" data-language-autonym="吴语" data-language-local-name="wu-kiina" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%90%D7%98%D7%A2%D7%A0%D7%A5" title="פאטענץ — jiddiš" lang="yi" hreflang="yi" data-title="פאטענץ" data-language-autonym="ייִדיש" data-language-local-name="jiddiš" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%AC%A1%E6%96%B9" title="次方 — kantoninkiina" lang="yue" hreflang="yue" data-title="次方" data-language-autonym="粵語" data-language-local-name="kantoninkiina" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%86%AA" title="冪 — kiina" lang="zh" hreflang="zh" data-title="冪" data-language-autonym="中文" data-language-local-name="kiina" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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