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338 lines
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<meta name="description" content="The exponent of a number says how many times to use the number in a multiplication">
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Exponents</h1>
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<p class="center">The <b>exponent</b> of a number says <b>how many times</b> to use the number in a multiplication.</p>
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<p> </p>
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<img src="algebra/images/exponent-8-2.svg" alt="8 to the Power 2" style="float:right; margin: 0 0 5px 20px;" height="132" width="143">
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<p class="center larger">In <b>8<sup>2</sup></b> the "2" says to use 8 twice in a multiplication,<br>
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so <b>8<sup>2</sup> = 8 × 8 = 64</b></p>
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<p>In words: 8<sup>2</sup> could be called "8 to the power 2" or "8 to the second power", or
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simply "8 squared"</p>
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<p class="center"><i>Exponents are also called Powers or Indices.</i></p>
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<p>Some more examples:</p>
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<div class="example">
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<h3 class="large">Example: <b>5<sup>3</sup> = 5 × 5 × 5 = 125</b></h3>
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<ul>
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<li>In words: 5<sup>3</sup> could be called "5 to the third power", "5 to the power 3" or simply
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"5 cubed"</li>
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</ul>
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</div>
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<div class="example">
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<h3 class="large">Example: <b>2<sup>4</sup> = 2 × 2 × 2 × 2 = 16</b></h3>
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<ul>
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<li>In words: 2<sup>4</sup> could be called "2 to the fourth power" or "2 to the power 4" or simply
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"2 to the 4th"</li>
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</ul>
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</div><p> </p>
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<p>Exponents make it easier to write and use many multiplications</p>
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<div class="example">
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<p>Example: <b>9<sup>6</sup></b> is easier to write and read than <b>9 × 9 × 9 × 9 × 9 × 9</b></p>
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</div>
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<p class="center larger">You can multiply <b>any</b> number by itself <b>as many times</b> as you want using exponents.</p>
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<p>Try here:</p>
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<div class="script" style="height: 200px;">
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algebra/images/exponent-calc.js
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</div>
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<p class="large">So<b> in general</b>:</p>
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<table align="center" width="85%" border="0">
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<tbody>
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<tr>
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<td style="text-align:right;"><span class="large">a<sup>n</sup></span> tells you to multiply <b>a</b> by itself,<br>
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so there are <b>n</b> of those <b>a</b>'s:</td>
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<td style="text-align:right;"> </td>
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<td><img src="algebra/images/exponent-definition.gif" alt="exponent definition" height="68" width="197"></td>
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</tr>
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</tbody></table>
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<h2>Another Way of Writing It</h2>
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<p>Sometimes people use the <b>^</b> symbol (above the 6 on your keyboard), as it is easy to type.</p>
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<div class="example">
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<p class="large">Example: <b>2^4 is the same as 2<sup>4</sup></b></p>
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<ul>
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<li><span class="large"><b>2^4 = 2 × 2 × 2 × 2 = 16</b></span></li>
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</ul>
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</div>
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<h2>Negative Exponents</h2>
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<p class="center">Negative? What could be the opposite of multiplying? <span class="large">Dividing! </span></p>
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<p>So we divide by the number each time, which is the same as multiplying by <span class="intbl"><em>1</em><strong>number</strong></span></p>
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<div class="example">
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<p class="large">Example: <b>8<sup>-1</sup> =</b> <b><span class="intbl"><em>1</em><strong>8</strong></span> = 0.125</b></p>
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</div>
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<p>We can continue on like this:</p>
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<div class="example">
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<p class="large">Example: <b>5<sup>-3</sup> = <span class="intbl"><em>1</em><strong>5</strong></span> × <span class="intbl"><em>1</em><strong>5</strong></span> × <span class="intbl"><em>1</em><strong>5</strong></span> = 0.008</b></p>
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</div>
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<p>But it is often easier to do it this way:</p>
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<div class="example">
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<p class="large"><b>5<sup>-3</sup></b> could also be calculated like:</p>
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<p class="center larger"><span class="intbl"><em>1</em><strong>5 × 5 × 5</strong></span> = <span class="intbl"><em>1</em><strong>5<sup>3</sup></strong></span> = <span class="intbl"><em>1</em><strong>125</strong></span> = 0.008</p>
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</div>
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<h2>Negative? Flip the Positive!</h2>
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<table width="100%" border="0">
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<tbody>
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<tr>
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<td><img src="algebra/images/negative-exponent.gif" alt="negative-exponent" height="57" width="99"></td>
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<td>
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<p>That last example showed an easier way to handle negative exponents:</p>
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<ul>
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<li>Calculate the positive exponent (<span class="large">a<sup>n</sup></span>)</li>
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<li>Then take the <a href="reciprocal.html">Reciprocal</a> (i.e. <span class="large">1/a<sup>n</sup></span>)</li>
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</ul></td>
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</tr>
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</tbody></table>
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<p>More Examples:</p>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th>Negative Exponent</th>
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<th> </th>
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<th>Reciprocal of<br>
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Positive Exponent</th>
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<th> </th>
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<th>Answer</th>
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</tr>
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<tr style="text-align:center;">
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<td><b>4<sup>-2</sup></b></td>
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<td> = </td>
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<td><b>1 / 4<sup>2</sup></b> </td>
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<td> = </td>
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<td><b>1/16 = 0.0625</b></td>
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</tr>
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<tr style="text-align:center;">
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<td><b>10<sup>-3</sup></b></td>
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<td> = </td>
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<td><b>1 / 10<sup>3</sup></b> </td>
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<td> = </td>
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<td><b>1/1,000 = 0.001</b></td>
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</tr>
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<tr style="text-align:center;">
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<td><b>(-2)<sup>-3</sup></b></td>
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<td> = </td>
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<td><b>1 / (-2)<sup>3</sup></b> </td>
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<td> = </td>
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<td><b>1/(-8) = -0.125</b></td>
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</tr>
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</tbody></table>
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</div>
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<h2>What if the Exponent is 1, or 0?</h2>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td class="huge">1</td>
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<td> </td>
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<td>If the exponent is 1, then you just have the number itself (example <b>9<sup>1</sup></b> = <b>9</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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<td> </td>
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</tr>
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<tr>
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<td class="huge">0</td>
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<td> </td>
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<td>If the exponent is 0, then you get <b>1</b> (example <b>9<sup>0</sup></b> = <b>1</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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<td> </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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<td>But what about <b>0<sup>0</sup></b> ? It could be either 1 or 0, and so people say it is <i>"indeterminate"</i>.</td>
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</tr>
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</tbody></table>
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<h2>It All Makes Sense</h2>
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<p>If you look at that table, you will see that positive, zero or
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negative exponents are really part of the same (fairly simple) pattern:</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="algebra/images/larger-smaller-5.svg" alt="exponent 5 times larger or 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%">5 × 5</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%">5</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%">1</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%"><span class="intbl"><em>1</em><strong>5</strong></span> <span class="intbl"><strong></strong></span></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%"><span class="intbl"><em>1</em><strong>5</strong></span> × <span class="intbl"><em>1</em><strong>5</strong></span></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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<h2>Be Careful About Grouping</h2>
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<p>To avoid confusion, use parentheses <span class="large">()</span> in cases like this:</p>
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<div class="simple">
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<table align="center" width="80%" border="0">
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<tbody>
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<tr>
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<td style="text-align:right;">With () :</td>
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<td class="large" align="center"><b>(−2)<sup>2</sup></b> = (−2) × (−2) = <b>4</b></td>
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</tr>
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<tr>
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<td style="text-align:right;">Without () :</td>
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<td class="large" align="center"><b>−2<sup>2</sup></b> = −(2<sup>2</sup>) = −(2 × 2) = <b>−4</b></td>
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</tr>
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</tbody></table>
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</div><br>
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<div class="simple">
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<table align="center" width="80%" border="0">
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<tbody>
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<tr>
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<td style="text-align:right;">With () :</td>
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<td class="large" align="center"><b>(ab)<sup>2</sup></b> = <b>ab × ab</b></td>
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</tr>
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<tr>
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<td style="text-align:right;">Without () :</td>
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<td class="large" align="center"><b>ab<sup>2</sup></b> = a × (b)<sup>2</sup> = <b>a × b × b</b></td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<div class="questions">305, 1679, 306, 1680, 1077, 1681, 1078, 1079, 3863, 3864</div>
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<div class="related">
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<a href="algebra/exponent-laws.html">Laws of Exponents</a>
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<a href="algebra/exponent-fractional.html">Fractional Exponents</a>
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<a href="index-notation-powers.html">Powers of 10</a>
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<a href="decimals.html">Decimals</a>
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<a href="metric-numbers.html">Metric Numbers</a>
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</div>
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<!-- #EndEditable -->
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</article>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2021 MathsIsFun.com</div>
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