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171 lines
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HTML
171 lines
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<h1 class="center">Exponents of Negative Numbers</h1>
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<h2>Squaring Removes Any Negative</h2>
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<p>"Squaring" means to multiply a number by itself.</p>
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<ul>
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<li>Squaring a <b>positive</b> number gets a <b>positive</b> result: (+5) × (+5) = +25</li>
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<li>Squaring a <b>negative</b> number also gets a <b>positive</b> result: (−5) × (−5) = +25</li>
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</ul>
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<p>Because a <a href="../multiplying-negatives.html">negative times a negative gives a positive</a>. So:</p>
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<div class="simple">
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<p class="center"><img src="images/square-root-pos-neg.svg" alt="5x5 = -5x-5" height="69" width="308"></p>
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<p><i class="larger">"So what?"</i> you say ...</p>
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<div class="indent50px">
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<p>... well take a look at this:</p>
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</div>
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<p class="center"><img src="images/exponent-sqroot-sqr.gif" alt="Square root of square" height="42" width="130"></p>
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<p>Oh no! We started with <b>minus 3</b> and ended with <b>plus 3</b>.</p>
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<p>When we <b>square</b> a number, then take the <b>square root</b>, we may not end up with the number we started with!</p>
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<p>In fact we end up with the <a href="../numbers/absolute-value.html">absolute value</a> of the number:</p>
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</div>
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<p class="center"><span class="larger"><b>√(x<b><sup>2</sup></b>)</b> = <b>|x|</b></span></p>
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<p> </p>
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<p>That also happens for all even (but not odd) <a href="../exponent.html">Exponents</a>.</p>
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<p>Try here:</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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<h2>Even Exponents of Negative Numbers</h2>
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<div class="def">
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<p>An even exponent always gives a <b>positive</b> (or 0) result.</p>
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</div>
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<p>That simple fact can make our life easier:</p>
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<div class="tbl">
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<div class="row"><span class="left">1 (Odd):</span><span class="right">(−1)<sup>1</sup> =<b> −1</b></span></div>
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<div class="row"><span class="left">2 (Even):</span><span class="right">(−1)<sup>2</sup> = (−1) × (−1) = <b>+1</b></span></div>
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<div class="row"><span class="left">3 (Odd):</span><span class="right">(−1)<sup>3</sup> = (−1) × (−1) × (−1) = <b>−1</b></span></div>
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<div class="row"><span class="left">4 (Even):</span><span class="right">(−1)<sup>4</sup> = (−1) × (−1) × (−1) × (−1) = <b>+1</b></span></div>
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</div>
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<p>Do you see the −1, +1, −1, +1 pattern?</p>
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<div class="def">
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<p><span class="larger">(−1)<sup>odd</sup></span> <span class="larger">=<b> −1</b></span></p>
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<p><span class="larger">(−1)<sup>even</sup></span> <span class="larger">=<b> +1</b></span></p>
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</div>
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<p>So we can "shortcut" some calculations, like:</p>
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<div class="example">
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<h3>Example: What is (−1)<sup>97</sup> ?</h3>
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<p>97 is odd, so:</p>
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<p class="center"><span class="larger">(−1)<sup>97</sup> = −1</span></p>
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</div>
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<div class="example">
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<h3>Example: What is (−2)<sup>6</sup> ?</h3>
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<p>2<sup>6</sup> = 64, and 6 is even, so:</p>
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<p class="center"><span class="larger">(−2)<sup>6</sup> = +64</span></p>
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</div>
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<h2>Roots of Negative Numbers</h2>
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<div class="example">
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<h3>Example: What is the value of x here: <span class="large">x<sup>2</sup> = −1</span></h3>
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<p><b>Does x=1?</b></p>
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<p class="center">1 × 1 = <b>+1</b></p>
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<p><b>Does x=−1?</b></p>
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<p class="center">(−1) × (−1) = <b>+1</b></p>
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<p>We can't get −1 for an answer!</p>
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</div>
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<p><b>It seems impossible!</b></p>
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<p>Well, it <b>is</b> impossible using <a href="../numbers/real-numbers.html">Real Numbers</a>.</p>
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<p>But we <b>can</b> do it using <a href="../numbers/imaginary-numbers.html">Imaginary Numbers</a>.</p>
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<p>In other words:</p>
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<p class="center large">√−1 is <b>not</b> a Real Number ...</p>
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<p class="center large">... it is an Imaginary Number</p>
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<p>This is true for <b>all even roots</b>:</p>
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<div class="def">
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<p>An Even Root of a Negative Number is Not Real</p>
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</div>
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<p>So just be careful when taking square roots, 4th roots, 6th roots, etc.</p>
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<p> </p>
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<div class="questions">1742, 3998, 459, 3999, 460, 1743, 1093, 4000, 1094, 4001</div>
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<div class="related">
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<a href="../exponent.html">Exponents</a>
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<a href="../multiplying-negatives.html">Negative Times a Negative Gives a Positive</a>
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