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325 lines
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HTML
325 lines
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<title>Squares and Square Roots in Algebra</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Squares and Square Roots in Algebra</h1>
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<p class="center">You might like to read our <a href="../square-root.html">Introduction to Squares and Square Roots</a> first.</p>
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<h2>Squares</h2>
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<div align="center">
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<p> <span class="larger">To square a number, just multiply it by itself
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... </span></p>
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</div>
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<div class="example">
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<h3>Example: What is 3 squared?</h3>
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<table align="center" cellpadding="6" border="0">
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<tbody>
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<tr valign="middle">
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<td class="larger">3 Squared</td>
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<td class="larger">=</td>
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<td><img src="images/3-squared-box.svg" alt="3x3 box"></td>
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<td class="larger">= 3 × 3 = <b>9</b></td>
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</tr>
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</tbody></table>
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</div>
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<p>"Squared" is often written as a little 2 like this:</p>
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<p align="center"><img src="images/4-squared.svg" alt="4 Squared equals 16"><br>
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This says <i><b>"4 Squared equals 16"</b></i><span class="Larger"><br>
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</span>(the little 2 means
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the number appears twice in multiplying, so <b>4×4</b>=16)</p>
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<h2>Square Root</h2>
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<p class="Larger">A <b>square root</b> goes the other direction:</p>
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<p align="center"><img src="images/square-vs-square-root.svg" alt="square root of 9 is 3"></p>
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<p align="center"><span class="Larger">3 squared is 9, so a <b>square root
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of 9 is 3</b></span></p>
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<p>It is like asking:</p>
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<div class="center80">
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<p class="large" align="center">What can I multiply by itself to get this?</p>
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</div>
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<h2>Definition</h2>
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<p>Here is the definition:</p>
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<div class="def">
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<p>A square root of x is <b>a number r</b> whose square is x:</p>
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<p align="center"><span class="large">r<sup>2</sup> = x</span><br>
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r is a square root of x</p>
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</div>
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<h2>The Square Root Symbol<br>
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</h2>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td align="center"><img src="../images/square-root-symbol.gif" alt="radical symbol" height="56" width="31"></td>
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<td> </td>
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<td>
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<p>This is the special symbol that means "square root",
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it is like a tick, <br>
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and actually started hundreds of years
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ago as a dot with a flick upwards.<br>
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<br>
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It is called the <i><b>radical</b></i>, and always makes mathematics look important!</p></td>
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</tr>
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</tbody></table>
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<p>We can use it like this:</p>
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<p align="center"> <img src="../images/square-root-9.gif" alt="square root of 9" align="absmiddle" height="33" width="92"> <br>
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we say "square root of 9 equals 3"</p>
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<div class="example">
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<h3>Example: What is √36 ?</h3>
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<p>Answer: 6 × 6 = 36, so <b>√36 = 6</b></p>
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</div>
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<h2>Negative Numbers</h2>
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<p>We can also square negative numbers.</p>
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<div class="example">
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<h3>Example: What is <b>minus 5 squared</b>? </h3>
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</div>
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<div class="center80">
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<p>But hang on ... what does "minus 5 squared" mean? </p>
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<ul>
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<li>square the 5, then do the minus?</li>
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<li>or square (−5)? </li>
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</ul>
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<p>It isn't clear! And we get different answers:</p>
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<ul>
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<li>square the 5, then do the minus: −(5×5) = <span class="hilite">−25</span></li>
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<li>square (−5): (−5)×(−5) = <span class="hilite">+25</span></li>
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</ul>
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<p>So let's make it clear by using "( )". </p>
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</div>
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<div class="example">
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<h3>Example Corrected: What is <b>(−5)<sup>2</sup></b> ?</h3>
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<p>Answer: </p>
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<p class="larger" align="center">(−5) × (−5) = <b>25 </b></p>
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<p align="center">(because a <a href="../multiplying-negatives.html">negative times a negative gives a positive</a>)</p>
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</div>
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<p>That was interesting!</p>
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<p class="larger" align="center">When we square a <b>negative</b> number we get a <b>positive</b> result. </p>
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<p>Just the same as when we square a positive number:</p>
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<p align="center"><img src="images/square-root-pos-neg.svg" alt="5x5 = -5x-5"></p>
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<p>Now remember our definition of a square root? </p>
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<div class="def">
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<p>A square root of x is <b>a number r</b> whose square is x:</p>
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<p align="center"><span class="large">r<sup>2</sup> = x</span><br>
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r is a square root of x</p>
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</div> <p>And we just found that:</p>
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<p class="center large">(+5)<sup>2</sup> = 25<br>
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(−5)<sup>2</sup> = 25</p>
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<p>So <b>both <span class="larger">+5 and −5</span></b> are square roots of 25</p>
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<h2>Two Square Roots</h2>
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<p>There can be a <b>positive</b> and <b>negative</b> square root!</p>
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<p>This is important to remember.</p>
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<div class="example">
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<h3>Example: Solve w<sup>2</sup> = a</h3>
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<p>Answer: </p>
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<p class="larger" align="center"><b>w = √a</b> and <b>w = −√a</b></p>
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</div>
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<h2>Principal Square Root</h2>
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<p>So if there are really two square roots, why do people say √<span class="larger">25</span> = <span class="larger">5 ?</span></p>
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<p class="larger" align="center">Because <b>√</b> means the <b>principal square root</b> ... the one that isn't negative!</p>
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<p>There <b>are</b> two square roots, but <b>the symbol <span class="large">√</span></b> means <b>just the principal square root</b>.</p>
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<div class="example">
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<h3>Example: </h3>
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<p align="center">The square roots of 36 are <span class="larger">6 <b>and</b> −6</span> </p>
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<p align="center">But<span class="larger"> </span>√<span class="larger">36</span> = <span class="larger">6</span> (not −6)</p>
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</div>
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<p>The Principal Square Root is sometimes called the Positive Square Root (but it can be zero).</p>
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<h2>Plus-Minus Sign</h2>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><span class="huge">±</span></td>
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<td> is a special symbol that means "plus or minus", </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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</tbody></table>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td align="right">so instead of writing:</td>
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<td> </td>
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<td><span class="larger"><b>w = √a</b> and <b>w = −√a</b></span></td>
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</tr>
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<tr>
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<td align="right">we can write:</td>
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<td> </td>
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<td><span class="large">w = ±√a</span></td>
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</tr>
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</tbody></table>
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<h2>In a Nutshell</h2>
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<div class="tbl">
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<div class="row"><span class="left">When we have:</span><span class="right"><span class="largest">r<sup>2</sup> = x</span></span></div>
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<div class="row"><span class="left">then:</span><span class="right"><span class="largest">r = ±√x</span></span></div>
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</div>
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<h2>Why Is This Important?</h2>
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<p>Why is this "plus or minus" important? Because we don't want to miss a solution!</p>
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<div class="example">
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<h3>Example: Solve x<sup>2</sup> − 9 = 0 </h3>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="large">x<sup>2</sup> − 9 = 0</span></span></div>
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<div class="row"><span class="left">Move 9 to right:</span><span class="right"><span class="large">x<sup>2</sup> = 9</span></span></div>
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<div class="row"><span class="left">Square Roots:</span><span class="right"><span class="large">x = ±√9</span></span></div>
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<div class="row"><span class="left">Answer:</span><span class="right"><span class="large">x = ±3 </span></span></div>
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</div>
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<p class="larger" align="center">The "<span class="large">±</span>" tells us to include the "−3" answer also.</p>
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<p class="center"><img src="images/graph-x2-9.gif" alt="x^2-9" height="164" width="153"></p>
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</div>
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<div class="example">
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<h3>Example: Solve for x in (x − 3)<sup>2</sup> = 16</h3>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="large">(x − 3)<sup>2</sup> = 16</span></span></div>
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<div class="row"><span class="left">Square Roots:</span><span class="right"><span class="large">x − 3 = ±√16</span></span></div>
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<div class="row"><span class="left">Calculate √16:</span><span class="right"><span class="large">x − 3 = </span> <span class="large">±4</span></span></div>
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<div class="row"><span class="left">Add 3 to both sides:</span><span class="right"><span class="large">x = 3 ± 4</span></span></div>
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<div class="row"><span class="left">Answer:</span><span class="right"><span class="large">x = 7 or −1</span></span></div>
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</div>
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<p>Check: (7−3)<sup>2</sup> = 4<sup>2</sup> = 16<br>
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Check: (−1−3)<sup>2</sup> = (−4)<sup>2</sup> = 16</p>
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</div>
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<h2>Square Root of xy</h2>
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<p>When two numbers are multiplied <b>within</b> a square root, we can split it into a multiplication of two square roots like this:</p>
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<div class="center80">
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<p class="center" style="font-size:150%;">√<span style="text-decoration: overline">xy</span> = √<span style="text-decoration: overline">x</span>√<span style="text-decoration: overline">y</span></p>
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<p align="center">but only when <b>x</b> and <b>y</b> are <b>both greater than or equal to 0</b></p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: What is <b>√(100×4)</b> ?</h3>
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<div class="tbl">
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<div class="row"><span class="left">√(100×4)</span><span class="right">= √(100) × √(4)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= 10 × 2</span></div>
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<div class="row"><span class="left"> </span><span class="right">= 20</span></div>
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</div>
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</div>
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<p>And <b>√<span style="text-decoration: overline">x</span>√<span style="text-decoration: overline">y</span> = √<span style="text-decoration: overline">xy</span></b> :</p>
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<div class="example">
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<h3>Example: What is <b>√8√2</b> ?</h3>
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<div class="tbl">
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<div class="row"><span class="left">√8√2</span><span class="right">= √(8×2)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= √16</span></div>
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<div class="row"><span class="left"> </span><span class="right">= 4</span></div>
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</div>
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</div>
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<div class="example">
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<h3>Example: What is <b>√(−8 × −2)</b> ?</h3>
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<div class="tbl">
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<div class="row"><span class="left">√(−8 × −2)</span><span class="right"> = √(−8) × √(−2)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= <b>???</b></span></div>
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</div>
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<p>We seem to have fallen into some trap here!</p>
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<p>We can use <a href="../numbers/imaginary-numbers.html">Imaginary Numbers</a>,
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but that leads to a <b>wrong</b> answer of <b>−4</b></p>
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<p>Oh that's right ... </p>
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<p class="larger" align="center">The rule only works when <b>x</b> and <b>y</b> are both greater than or equal to 0</p>
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<p>So we can't use that rule here.</p>
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<p>Instead just do it this way: </p>
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<p class="center large">√(−8 × −2) = √16 = +4</p>
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</div>
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<div class="fun">
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<h3>Why does √<span style="text-decoration: overline">xy</span> = √<span style="text-decoration: overline">x</span>√<span style="text-decoration: overline">y</span> ?</h3>
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<p>We can use the fact that squaring a square root gives us the original value back again:</p>
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<p class="center larger">(√<span style="text-decoration: overline">a</span>)<sup>2</sup> = a</p>
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<p>Assuming <b>a</b> is not negative!</p>
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<div class="tbl">
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<div class="row"><span class="left">We can do that for xy:</span><span class="right">(√<span style="text-decoration: overline">xy</span>)<sup>2</sup> = xy</span></div>
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<div class="row"><span class="left">And also to x, and y, separately:</span><span class="right">(√<span style="text-decoration: overline">xy</span>)<sup>2</sup> = (√<span style="text-decoration: overline">x</span>)<sup>2</sup>(√<span style="text-decoration: overline">y</span>)<sup>2</sup></span></div>
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<div class="row"><span class="left">Use a<sup>2</sup>b<sup>2</sup> = (ab)<sup>2</sup>:</span><span class="right">(√<span style="text-decoration: overline">xy</span>)<sup>2</sup> = (√<span style="text-decoration: overline">x</span>√<span style="text-decoration: overline">y</span>)<sup>2</sup></span></div>
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<div class="row"><span class="left">Remove square from both sides<sup></sup>:</span><span class="right">√<span style="text-decoration: overline">xy</span> = √<span style="text-decoration: overline">x</span>√<span style="text-decoration: overline">y</span></span></div>
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</div>
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</div>
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<h2>An Exponent of a Half </h2>
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||
<p>A square root can also be written as a <a href="exponent-fractional.html">fractional exponent</a> of one-half:</p>
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||
<p align="center"><img src="images/square-root-exponent-half.gif" alt="square-root-exponent-half" height="33" width="110"><br>
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but only for <b>x</b> greater than or equal to 0</p>
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||
<h2>How About the Square Root of Negatives?</h2>
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||
<p>The result is an <a href="../numbers/imaginary-numbers.html">Imaginary Number</a>... read that page to learn more.</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(457, 458, 1084, 1085, 1086, 2286, 2287, 3994, 3995, 3996);</script> <a href="javascript:doQ(173)">Harder Question</a>
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</div>
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<div class="related"> <a href="../irrational-numbers.html">Irrational Numbers</a> <a href="../surds.html">Surds</a> <a href="../scientific-calculator.html">Scientific Calculator</a> <a href="index.html">Algebra Index</a> </div>
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