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130 lines
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HTML
130 lines
6.9 KiB
HTML
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<title>Using Exponents in Algebra</title>
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<h1 align="center">Using Exponents in Algebra</h1>
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<p>You might like to read the page on <a href="../exponent.html">Exponents</a> first.</p>
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<h2>Whole Number Exponents</h2>
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<table border="0" align="center">
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<tr>
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<td align="right"><p>The exponent "n" in <span class="large">a<sup>n</sup></span> says <b>how many times </b>to use <span class="large">a</span> in a multiplication:</p></td>
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<td align="right"> </td>
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<td><img src="images/exponent-definition.gif" alt="exponent definition" width="197" height="68" /></td>
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</tr>
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</table>
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<br />
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<div class="example">
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<h3 class="Larger">Example: <b>5<sup>3</sup> = 5 × 5 × 5 = 125</b></h3>
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<ul>
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<li>The "3" says to use 5 three times in a multiplication</li>
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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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<h2>Negative Exponents</h2>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="images/negative-exponent.gif" alt="negative-exponent" width="99" height="57" /></p>
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<p>A negative exponent <span class="large">a<sup>−n</sup></span> is equal to <span class="large">1/a<sup>n</sup></span> (1 divided by the positive exponent)</p><div style="clear:both"></div>
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<div class="example">
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<h3>Example: </h3>
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<p class="center larger"><b>5<sup>−3</sup> = </b><b> 1/5<sup>3</sup> = 1/125 = 0.008</b> </p>
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</div>
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<p>Also ... by changing the signs of the exponents we get:</p>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="images/negative-exponent-2.gif" alt="negative-exponent" width="112" height="61" /></p>
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<p>A positive exponent <span class="large">a<sup>n</sup></span> is equal to <span class="large">1/a<sup>−n</sup></span> (1 divided by the negative exponent)</p>
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<div style="clear:both"></div>
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<p>So, we can move an expression between the top and bottom (numerator and denominator) of a fraction by simply changing the sign of
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the exponent.</p>
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<div class="example">
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<p class="larger"><span class="Larger">Example: <b>x<sup>−1</sup> = </b></span><b> 1/x</b> (a simple <a href="../reciprocal.html">reciprocal</a>) </p>
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</div>
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<div class="example">
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<p class="larger"><span class="Larger">Example: <b>5<sup>−3</sup> = </b></span><b> 1/5<sup>3</sup> = 1/125 = 0.008</b> </p>
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</div>
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<h2>Positive and Negative Together</h2>
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<p>Here is an example with positive and negative exponents:</p>
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<div class="example">
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<h3>Example: 4<sup>3</sup>2<sup>-5</sup>9<sup>-1</sup>3<sup>2</sup></h3>
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<p>We can put the negative exponents at the bottom (and they become positive):</p>
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<p class="center large"><span class="intbl"><em>4<sup>3</sup>3<sup>2</sup></em><strong>2<sup>5</sup>9</strong></span></p>
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<p>3<sup>2</sup> is 9:</p>
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<p class="center large"><span class="intbl"><em>4<sup>3</sup>9</em><strong>2<sup>5</sup>9</strong></span></p>
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<p>The 9s cancel out:</p>
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<p class="center large"><span class="intbl"><em>4<sup>3</sup></em><strong>2<sup>5</sup></strong></span></p>
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<p><b>4<sup>3</sup></b> is 4 × 4 × 4 = 64, and <strong>2<sup>5</sup></strong> is 2 × 2 × 2 × 2 × 2 = 32<strong><sup></sup></strong>:</p>
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<p class="center large"><span class="intbl"><em>64</em><strong>32</strong></span></p>
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<p>Which simplifies to:</p>
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<p class="center large">2</p>
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<p>Done! </p>
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</div>
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<p> </p>
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<script type="text/javascript">getQ(455, 456, 1080, 1081, 2058, 2059, 2060, 2061, 3870, 3871);</script>
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<div class="related"> <a href="../exponent.html">Exponents</a> <a href="exponent-laws.html">Laws of Exponents</a> <a href="exponent-fractional.html">Fractional Exponents</a> <a href="exponents-squaring-negative.html">Exponents of Negative Numbers</a> </div>
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