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<title>Fractional Exponents</title>
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<h1 class="center">Fractional Exponents</h1>
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<p class="center">Also called <b>"Radicals"</b> or <b>"Rational Exponents"</b></p>
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<h2>Whole Number Exponents</h2>
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<div class="simple">
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<p>First, let us look at whole number <a href="../exponent.html">exponents</a>:</p>
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<p style="float:left; margin: 20px 30px 5px 0;"><img src="images/exponent-8-2.svg" alt="8 to the Power 2"></p>
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<p>The exponent of a number says <b>how many times</b> to use the number in a <b>multiplication.</b></p>
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<p class="larger">In this example: <b>8<sup>2</sup> = 8 × 8 = 64</b></p>
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<div style="clear:both"></div>
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<div class="words">In words: 8<sup>2</sup> could be called "8 to the second power", "8 to the power 2" or
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simply "8 squared"</div>
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</div>
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<div class="example">
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<p>Another example: <span class="larger"><b>5<sup>3</sup> = 5 × 5 × 5 = 125</b></span></p>
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</div>
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<h2>Fractional Exponents</h2>
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<p>But what if the exponent is a fraction?</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>
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<p>An exponent of <span class="intbl"><em>1</em><strong>2</strong></span> is a <b>square root</b></p>
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<p>An exponent of <span class="intbl"><em>1</em><strong>3</strong></span> is a <b>cube root</b></p>
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<p>An exponent of <span class="intbl"><em>1</em><strong>4</strong></span> is a <b>4th root</b></p>
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<p>And so on!</p></td>
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<td style="width:20px;"> </td>
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<td valign="top"><img src="images/exponent-fractional-1.svg" alt="fractional exponents: 4^(1/2) = square root of 4, etc"></td>
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</tr>
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</tbody></table>
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<h2>Why?</h2>
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<p>Let's see why in an example.</p>
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<p>First, the <a href="exponent-laws.html">Laws of Exponents</a> tell us how to handle exponents when we multiply:</p>
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<div class="example">
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<h3>Example: x<sup>2</sup>x<sup>3</sup> = (xx)(xxx) = xxxxx = x<sup>5</sup></h3>
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<p>Which shows that <b>x<sup>2</sup>x<sup>3</sup> = x<sup>(2+3)</sup> = x<sup>5</sup></b></p>
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</div>
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<p>So let us try that with fractional exponents:</p>
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<div class="example">
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<h3>Example: What is 9<sup>½</sup> × 9<sup>½</sup> ?</h3>
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<p class="center"><span class="larger">9<sup>½</sup> × 9<sup>½</sup> = 9<sup>(½+½)</sup> = 9<sup>(1)</sup> = 9</span></p>
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<p>So <span class="larger">9<sup>½</sup></span> times itself gives 9.</p>
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<p>Now, what do we call a number that, when multiplied by itself, gives another number? The <a href="../square-root.html">square root</a> of that other number!</p>
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<p>See:</p>
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<p class="center larger">√9 × √9 = 9</p>
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<p>And:</p>
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<p class="center"><span class="larger">9<sup>½</sup> × 9<sup>½</sup> = 9</span></p>
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<p>So <span class="larger">9<sup>½</sup></span> is the same as <span class="larger">√9</span></p>
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</div>
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<h2>Try Another Fraction</h2>
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<p>Let us try that again, but with an exponent of one-quarter (1/4):</p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center"><span class="larger">16<sup>¼</sup> × 16<sup>¼</sup> × 16<sup>¼</sup> × 16<sup>¼</sup> = 16<sup>(¼+¼+¼+¼)</sup> = 16<sup>(1)</sup> = 16</span></p>
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<p>So <span class="larger">16<sup>¼</sup></span> used 4 times in a multiplication gives 16,</p>
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<p class="center">and so<b> <span class="larger">16<sup>¼</sup></span> is a 4th root of 16</b></p>
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</div>
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<h2>General Rule</h2>
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<p>It worked for <b>½</b>, it worked with <b>¼</b>, in fact it works generally:</p>
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<p class="center larger">x<sup>1/<b>n</b></sup> = The <b>n-</b>th Root of x</p>
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<p>In other words:</p>
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<div class="def">
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A fractional exponent like <b>1/n</b> means to<b> take the n-th root</b>:
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<p class="center large">
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x<sup>1/n</sup> =
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">x</span>
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</p>
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</div>
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<div class="example">
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<h3>Example: What is 27<sup>1/3</sup> ?</h3>
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<p>Answer: 27<sup>1/3</sup> =
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">27</span>
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= 3</p>
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</div>
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<h2>What About More Complicated Fractions?</h2>
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<p>What about a fractional exponent like <span class="large">4<sup>3/2</sup></span> ?</p>
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<p class="center">That is really saying to do a <b>cube</b> (3) and a <b>square root</b> (1/2), in any order.</p>
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<p>Let me explain.</p>
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<p>A fraction (like <b>m/n</b>) can be broken into two parts:</p>
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<ul>
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<li>a whole number part (<b>m</b>) , and</li>
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<li>a fraction (<b>1/n</b>) part</li>
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</ul>
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<p>So, because <b>m/n = m × (1/n)</b> we can do this:</p>
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<p class="center large">
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x<sup>m/n</sup> =
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x<sup>(m × 1/n)</sup> =
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(x<sup>m</sup>)<sup>1/n</sup> =
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">x<sup>m</sup></span>
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</p>
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<p>The order does not matter, so it also works for <b>m/n = (1/n) × m</b>:</p>
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<p class="center large">
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x<sup>m/n</sup> =
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x<sup>(1/n × m)</sup> =
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(x<sup>1/n</sup>)<sup>m</sup> =
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(<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">x</span> )<sup>m</sup>
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</p><p></p>
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<p>And we get this:</p>
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<div class="def">
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A fractional exponent like <b>m/n</b> means:
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<div class="tbl">
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<div class="row">
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<span class="lt">Do the <b>m-th power</b>, then take the <b>n-th root</b>:</span>
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<span class="rt">
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x<sup>m/n</sup> =
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">x<sup>m</sup></span>
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</span>
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</div>
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<div class="row">or </div>
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<div class="row">
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<span class="lt">Take the<b> n-th root</b>, then do the <b>m-th power</b>:</span>
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<span class="rt">
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x<sup>m/n</sup> =
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(<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">x</span> )<sup>m</sup> </span>
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</div>
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</div>
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</div>
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<p> </p>
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<p>Some examples:</p>
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<div class="example">
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<h3>Example: What is 4<sup>3/2</sup> ?</h3>
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<p class="center large">4<sup>3/2</sup> = 4<sup>3×(1/2)</sup> = √(4<sup>3</sup>) = √(4×4×4) = √(64) = <b>8</b></p>
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<p class="center">or</p>
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<p class="center large">4<sup>3/2</sup> = 4<sup>(1/2)×3</sup> = (√4)<sup>3</sup> = (2)<sup>3</sup> = <b>8</b></p>
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<p>Either way gets the same result.</p>
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</div>
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<div class="example">
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<h3>Example: What is 27<sup>4/3</sup> ?</h3>
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<p class="center large">27<sup>4/3</sup> = 27<sup>4×(1/3)</sup>
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= <span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">27<sup>4</sup></span>
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= <span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">531441</span>
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= <b>81</b></p>
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<p class="center">or</p>
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<p class="center large">27<sup>4/3</sup> = 27<sup>(1/3)×4</sup>
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= (<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">27</span> )<sup>4</sup> = (3)<sup>4</sup> = <b>81</b></p>
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<p>It was certainly easier the 2nd way!</p>
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</div>
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<h2>Now ... Play With The Graph!</h2>
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<p>See how <i>smoothly</i> the curve changes when you play with the fractions in this animation, this shows you that this idea of fractional exponents fits together nicely:</p>
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<div class="script">images/graph-exponent.js</div>
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<p>Things to try:</p>
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<ul>
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<li>Start with m=1 and n=1, then slowly increase n so that you can see 1/2, 1/3 and 1/4</li>
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<li>Then try m=2 and slide n up and down to see fractions like 2/3 etc</li>
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<li>Now try to make the exponent −1</li>
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<li>Lastly try increasing m, then reducing n, then <i>reducing</i> m, then <i>increasing</i> n: the curve should go around and around</li>
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</ul><p> </p>
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<div class="questions">321,322, 1089, 1090, 1091, 1092, 2266, 3997, 17, 2267</div>
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<div class="related">
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<a href="exponent-laws.html">Laws of Exponents</a>
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<a href="../exponent.html">Exponent</a>
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<a href="../index-notation-powers.html">Powers of 10</a>
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<a href="index.html">Algebra Menu</a>
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