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513 lines
19 KiB
HTML
513 lines
19 KiB
HTML
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<title>Laws of Exponents</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Laws of Exponents</h1>
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<p class="center">Exponents are also called <b>Powers</b> or <b>Indices</b></p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/exponent-8-2.svg" alt="8 to the Power 2" height="132" width="143"></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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<p>Try it yourself:</p>
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<div class="script" style="height: 200px;">
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images/exponent-calc.js
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</div>
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<p class="center large">So an Exponent saves us writing out lots of multiplies!</p>
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<div class="example">
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<h3>Example: <span class="larger">a<b><sup>7</sup></b></span></h3>
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<p class="center larger">a<b><sup>7</sup></b> = a × a × a × a × a × a × a = aaaaaaa</p>
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</div>
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<p>Notice how we wrote the letters together to mean multiply? We will do that a lot here.</p>
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<div class="example">
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<h3>Example: x<sup>6</sup> = xxxxxx</h3>
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</div>
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<h2>The Key to the Laws</h2>
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<p class="center large">Writing all the letters down is the key to understanding the Laws</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>5</sup></b>, but more on that later!</p>
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</div>
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<p>So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.</p>
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<a id="ideas"></a>
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<h2>All you need to know ...</h2>
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<p>The "Laws of Exponents" (also called "Rules of Exponents") come from <b>three ideas</b>:</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><img src="../images/style/pencil-paper.gif" alt="pencil paper" height="35" width="39"></td>
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<td>The exponent says<b> how many times</b> to use the number in a multiplication<b>.</b></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td><img src="../images/style/turn-over.gif" alt="turn over" height="41" width="45"></td>
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<td>A <b>negative exponent</b> means<b> divide</b>, because the opposite of multiplying is dividing</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td><img src="../images/style/pie-slice.gif" alt="pie slice" height="42" width="43"></td>
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<td>
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<table width="100%" border="0">
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<tbody>
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<tr>
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<td width="75%">A <a href="exponent-fractional.html">fractional exponent</a> like <b>1/n</b> means to<b> take the <a href="../numbers/nth-root.html">nth root</a></b>:</td>
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<td width="25%">
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x<sup>(<span class="intbl"><em>1</em><strong>n</strong></span>)</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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</td>
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</tr>
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</tbody></table></td>
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</tr>
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</tbody></table><br>
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<p class="center large">If you understand those, then you understand exponents!</p>
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<p class="center">And all the laws below are based on those ideas.</p>
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<h2>Laws of Exponents</h2>
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<p>Here are the Laws
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(explanations follow):</p>
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<div class="beach">
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<table align="center" width="90%" cellspacing="3" border="0">
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<tbody>
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<tr style="text-align:center;">
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<th width="50%">Law</th>
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<th width="50%">Example</th>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">x<sup>1</sup> = x</span></td>
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<td width="50%">6<sup>1</sup> = 6</td>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">x<sup>0</sup> = 1</span></td>
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<td width="50%">7<sup>0</sup> = 1</td>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">x<sup>-1</sup> = 1/x</span></td>
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<td width="50%">4<sup>-1</sup> = 1/4</td>
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</tr>
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<tr style="text-align:center;">
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<td class="large" width="50%"><br>
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</td>
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<td width="50%"><br>
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</td>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">x<sup>m</sup>x<sup>n</sup> = x<sup>m+n</sup></span></td>
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<td width="50%">x<sup>2</sup>x<sup>3</sup> = x<sup>2+3</sup> = x<sup>5</sup></td>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">x<sup>m</sup>/x<sup>n</sup> = x<sup>m-n</sup></span></td>
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<td width="50%">x<sup>6</sup>/x<sup>2</sup> = x<sup>6-2</sup> = x<sup>4</sup></td>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">(x<sup>m</sup>)<sup>n</sup> = x<sup>mn</sup></span></td>
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<td width="50%">(x<sup>2</sup>)<sup>3</sup> = x<sup>2×3</sup> = x<sup>6</sup></td>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">(xy)<sup>n</sup> = x<sup>n</sup>y<sup>n</sup></span></td>
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<td width="50%">(xy)<sup>3</sup> = x<sup>3</sup>y<sup>3</sup></td>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">(x/y)<sup>n</sup> = x<sup>n</sup>/y<sup>n</sup></span></td>
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<td width="50%">(x/y)<sup>2</sup> = x<sup>2</sup> / y<sup>2</sup></td>
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</tr>
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<tr style="text-align:center;">
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<td width="50%"><span class="large">x<sup>-n</sup> = 1/x<sup>n</sup></span></td>
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<td width="50%">x<sup>-3</sup> = 1/x<sup>3</sup></td>
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</tr>
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<tr>
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<td colspan="2">And the law about Fractional Exponents:</td>
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</tr>
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<tr style="text-align:center;">
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<td class="large" width="50%">
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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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<br>
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=
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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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</td>
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<td class="large" width="50%">
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x<sup>2/3</sup> =
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">x<sup>2</sup></span>
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<br>
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=
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(<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">x</span> )<sup>2</sup>
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</td>
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</tr>
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</tbody></table>
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</div>
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<h2>Laws Explained</h2>
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<p>The first three laws above (<span class="large">x<sup>1</sup> = x</span>, <span class="large">x<sup>0</sup> = 1</span> and <span class="large">x<sup>-1</sup> = 1/x</span>) are just part of the natural sequence of exponents. Have a look at this:</p>
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<div class="simple">
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<table align="center" width="60%" border="0">
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<tbody>
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<tr style="text-align:center;">
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<th colspan="4">Example: Powers of 5</th>
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</tr>
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<tr>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">.. etc..</td>
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<td style="text-align:center;"> </td>
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<td rowspan="7" width="19%"><img src="images/larger-smaller-5.svg" alt="exponent 5x larger smaller" height="199" width="69"></td>
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</tr>
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<tr>
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<td width="11%"><b>5<sup>2</sup></b></td>
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<td width="41%"><b>1 × 5 × 5</b></td>
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<td width="29%">25</td>
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</tr>
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<tr>
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<td width="11%"><b>5<sup>1</sup></b></td>
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<td width="41%"><b>1 × 5</b></td>
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<td width="29%">5</td>
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</tr>
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<tr>
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<td width="11%"><b>5<sup>0</sup></b></td>
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<td width="41%"><b>1</b></td>
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<td width="29%">1</td>
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</tr>
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<tr>
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<td width="11%"><b>5<sup>-1</sup></b></td>
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<td width="41%"><b>1 ÷ 5</b></td>
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<td width="29%">0.2</td>
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</tr>
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<tr>
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<td width="11%"><b>5<sup>-2</sup></b></td>
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<td width="41%"><b>1 ÷ 5 ÷ 5</b></td>
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<td width="29%">0.04</td>
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</tr>
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<tr>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">.. etc..</td>
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<td style="text-align:center;"> </td>
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</tr>
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</tbody></table>
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</div>
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<p>Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).</p>
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<h2>The law that x<sup>m</sup>x<sup>n</sup> = x<sup>m+n</sup></h2>
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<p class="indent50px">With x<sup>m</sup>x<sup>n</sup>, how many times do we end up multiplying "x"? <i>Answer:</i> first "m" times, then <b>by another</b> "n" times, for a total of "m+n" times.</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 class="indent50px">So, x<sup>2</sup>x<sup>3</sup> = x<sup>(2+3)</sup> = x<sup>5</sup></p>
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</div>
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<h2>The law that x<sup>m</sup>/x<sup>n</sup> = x<sup>m-n</sup></h2>
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<p class="indent50px">Like the previous example, how many times do we end up multiplying "x"? Answer: "m" times, then <b>reduce that</b> by "n" times (because we are dividing), for a total of "m-n" times.</p>
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<div class="example">
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<h3>Example: x<sup>4</sup>/x<sup>2</sup> = (xxxx) / (xx) = xx = x<sup>2</sup></h3>
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<p class="indent50px">So, x<sup>4</sup>/x<sup>2</sup> = x<sup>(4-2)</sup> = x<sup>2</sup></p>
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</div>
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<p class="indent50px">(Remember that <b>x</b>/<b>x</b> = 1, so every time you see an <b>x</b> "above the line" and one "below the line" you can cancel them out.)</p>
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<p>This law can also show you why <b>x<sup>0</sup>=1</b> :</p>
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<div class="example">
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<h3>Example: x<sup>2</sup>/x<sup>2</sup> = <b><b>x<sup>2-2</sup></b> = <b>x<sup>0</sup></b> =1</b></h3>
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</div>
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<h2>The law that (x<sup>m</sup>)<sup>n</sup> = x<sup>mn</sup></h2>
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<p class="indent50px">First you multiply "m" times. Then you have <b>to do that "n" times</b>, for a total of m×n times.</p>
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<div class="example">
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<h3>Example: (x<sup>3</sup>)<sup>4</sup> = (xxx)<sup>4</sup> = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x<sup>12</sup></h3>
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<p class="indent50px">So (x<sup>3</sup>)<sup>4</sup> = x<sup>3×4</sup> = x<sup>12</sup></p>
|
||
</div>
|
||
|
||
|
||
<h2>The law that (xy)<sup>n</sup> = x<sup>n</sup>y<sup>n</sup></h2>
|
||
|
||
<p class="indent50px">To show how this one works, just think of re-arranging all the "x"s and "y"s as in this example:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: (xy)<sup>3</sup> = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x<sup>3</sup>y<sup>3</sup></h3>
|
||
</div>
|
||
|
||
|
||
<h2>The law that (x/y)<sup>n</sup> = x<sup>n</sup>/y<sup>n</sup></h2>
|
||
|
||
<p class="indent50px">Similar to the previous example, just re-arrange the "x"s and "y"s</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: (x/y)<sup>3</sup> = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x<sup>3</sup>/y<sup>3</sup></h3>
|
||
</div>
|
||
|
||
|
||
<h2>The law that
|
||
|
||
|
||
x<sup>m/n</sup> =
|
||
<span class="nthroot">n</span><span style="font-size:110%;">√</span><span class="overline">x<sup>m</sup></span>
|
||
=
|
||
(<span class="nthroot">n</span><span style="font-size:110%;">√</span><span class="overline">x</span> )<sup>m</sup>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</h2>
|
||
|
||
<p class="indent50px">OK, this one is a little more complicated!</p>
|
||
<p class="indent50px">I suggest you read <a href="exponent-fractional.html">Fractional Exponents</a> first, so this makes more sense.</p>
|
||
<p class="indent50px">Anyway, the important idea is that:</p>
|
||
<p class="indent50px" align="center"><span class="larger">x<sup>1/<b>n</b></sup> = The <b>n-</b>th Root of x</span></p>
|
||
<p class="indent50px">And so a fractional exponent like <span class="large">4<sup>3/2</sup></span> is really saying to do a <b>cube</b> (3) and a <b>square root</b> (1/2), in any order.</p>
|
||
<p class="indent50px">Just remember from fractions that <b>m/n = m × (1/n)</b>:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example:
|
||
|
||
x<sup>(<span class="intbl"><em>m</em><strong>n</strong></span>)</sup> =
|
||
|
||
x<sup>(m × <span class="intbl"><em>1</em><strong>n</strong></span>)</sup> =
|
||
(x<sup>m</sup>)<sup>1/n</sup> =
|
||
|
||
<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">x<sup>m</sup></span>
|
||
|
||
|
||
</h3>
|
||
</div>
|
||
<p class="indent50px">The order does not matter, so it also works for <b>m/n = (1/n) × m</b>:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example:
|
||
|
||
|
||
x<sup>(<span class="intbl"><em>m</em><strong>n</strong></span>)</sup> =
|
||
|
||
x<sup>(<span class="intbl"><em>1</em><strong>n</strong></span> × m)</sup> =
|
||
(x<sup>1/n</sup>)<sup>m</sup> =
|
||
|
||
(<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">x</span> )<sup>m</sup>
|
||
|
||
</h3>
|
||
</div>
|
||
|
||
|
||
<h2>Exponents of Exponents ...</h2>
|
||
|
||
<p>What about this example?</p>
|
||
<p class="largest center">4<sup>3<sup>2</sup></sup></p>
|
||
<p>We do the exponent at the <b>top first</b>, so we calculate it this way:</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;">Start with:</td>
|
||
<td> </td>
|
||
<td style="text-align:center;"><span class="large">4<sup>3<sup>2</sup></sup></span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">3<sup>2</sup> = 3×3:</td>
|
||
<td> </td>
|
||
<td style="text-align:center;"><span class="large">4<sup>9</sup></span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">4<sup>9</sup> = 4×4×4×4×4×4×4×4×4:</td>
|
||
<td> </td>
|
||
<td class="large" align="center">262144</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
|
||
|
||
<h2>And That Is It!</h2>
|
||
|
||
<p><i>If you find it hard to remember all these rules, then remember this:</i></p>
|
||
<p class="center larger">you can work them out when you understand the<br>
|
||
<a href="#ideas">three ideas</a> near the top of this page:</p>
|
||
<ul>
|
||
<li>The exponent says<b> how many times</b> to use the number in a multiplication</li>
|
||
<li>A <b>negative exponent</b> means<b> divide</b></li>
|
||
<li>A fractional exponent like <b>1/n</b> means to<b> take the nth root</b>:
|
||
x<sup>(<span class="intbl"><em>1</em><strong>n</strong></span>)</sup> =
|
||
<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">x</span>
|
||
</li></ul>
|
||
<p class="center"> </p>
|
||
|
||
<h3>Oh, One More Thing ... What if x = 0?</h3>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>Positive Exponent (n>0)</td>
|
||
<td style="width:20px;"> </td>
|
||
<td>0<sup>n</sup> = 0</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Negative Exponent (n<0)</td>
|
||
<td> </td>
|
||
<td>0<sup>-n</sup> is <b>undefined</b> (because <a href="../numbers/dividing-by-zero.html">dividing by 0</a> is undefined)</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Exponent = 0</td>
|
||
<td> </td>
|
||
<td>0<sup>0</sup> ... <i>ummm</i> ... see below!</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
<div class="fun">
|
||
|
||
<h3>The Strange Case of 0<sup>0</sup></h3>
|
||
<p>There are different arguments for the correct value of 0<sup>0</sup></p>
|
||
<p>0<sup>0</sup> could be 1, or possibly 0, so some people say it is really "indeterminate":</p>
|
||
|
||
<table align="center" width="80%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td rowspan="3" width="10%"><img src="../images/style/question.gif" alt="question mark" height="47" width="26"></td>
|
||
<td width="28%">x<sup>0</sup> = 1, so ...</td>
|
||
<td width="62%"> 0<sup>0</sup> = 1</td>
|
||
</tr>
|
||
<tr>
|
||
<td width="28%">0<sup>n</sup> = 0, so ...</td>
|
||
<td width="62%"> 0<sup>0</sup> = 0</td>
|
||
</tr>
|
||
<tr>
|
||
<td width="28%">When in doubt ...</td>
|
||
<td width="62%">0<sup>0</sup> = <i>"indeterminate"</i></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">323, 2215, 2306, 324, 2216, 2307, 371, 2217, 2308, 2309</div>
|
||
|
||
<div class="related">
|
||
<a href="../exponent.html">Exponent</a>
|
||
<a href="exponent-fractional.html">Fractional Exponents</a>
|
||
<a href="../index-notation-powers.html">Powers of 10</a>
|
||
<a href="index.html">Algebra Menu</a>
|
||
</div>
|
||
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