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335 lines
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335 lines
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HTML
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<div id="hdr"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Order of Operations<br>
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<i>GEMS</i></h1>
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<h3>Operations</h3>
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<p><i><b>"Operations"</b></i> mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.</p>
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<p class="center">But, when you see something like ...</p>
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<p class="center large">7 + (6 × 5<sup>2</sup> + 3)</p>
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<p class="center">... what part should you calculate first?<br>
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<br>
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Start at the left and go to the right?<br>
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Or go from right to left?</p>
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<p class="center"><i>Warning: Calculate them in the wrong order, and you can get a wrong answer !</i></p>
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<p>So, long ago people agreed to follow rules when doing calculations, and they are:</p>
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<h2>Order of Operations</h2>
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<p class="large">Inside the Grouping symbols () [] first</p>
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<table align="center" cellpadding="3" border="0">
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<tbody>
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<tr>
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<td class="large"><img src="images/style/yes.svg" alt="yes" height="30" width="30"></td>
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<td class="large"> </td>
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<td class="large">4 × <span class="hilite">(5 + 3)</span></td>
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<td class="large">=</td>
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<td class="large">4 × <span class="hilite">8</span></td>
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<td class="large">=</td>
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<td class="large">
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<div class="center"><b>32</b></div>
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</td>
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<td class="large"> </td>
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</tr>
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<tr>
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<td class="large"><img src="images/style/no.svg" alt="not" height="30" width="30"></td>
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<td class="large"> </td>
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<td class="large">4 × (5 + 3) </td>
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<td class="large">=</td>
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<td class="large"><span class="wrong">20</span> + 3</td>
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<td class="large">=</td>
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<td class="large">
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<div class="center">23</div>
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</td>
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<td class="large">(wrong)</td>
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</tr>
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</tbody></table><br>
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<p class="large"><a href="exponent.html">Exponents</a> (Powers, Roots) before Multiply, Divide, Add or Subtract</p>
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<table align="center" cellpadding="3" border="0">
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<tbody>
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<tr>
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<td class="large"><img src="images/style/yes.svg" alt="yes" height="30" width="30"></td>
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<td class="large"> </td>
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<td class="large">5 × <span class="hilite">2<sup>2</sup></span></td>
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<td class="large">=</td>
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<td class="large">5 × <span class="hilite">4</span></td>
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<td class="large">=</td>
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<td class="large">
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<div class="center"><b>20</b></div>
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</td>
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<td class="large"> </td>
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</tr>
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<tr>
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<td class="large"><img src="images/style/no.svg" alt="not" height="30" width="30"></td>
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<td class="large"> </td>
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<td class="large">5 × 2<sup>2</sup></td>
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<td class="large">=</td>
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<td class="large"><span class="wrong">10</span><sup>2</sup></td>
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<td class="large">=</td>
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<td class="large">
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<div class="center">100</div>
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</td>
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<td class="large">(wrong)</td>
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</tr>
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</tbody></table><br>
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<p class="large">Multiply or Divide before you Add or Subtract</p>
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<table align="center" cellpadding="3" border="0">
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<tbody>
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<tr>
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<td class="large"><img src="images/style/yes.svg" alt="yes" height="30" width="30"></td>
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<td class="large"> </td>
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<td class="large">2 + <span class="hilite">5 × 3</span></td>
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<td class="large">=</td>
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<td class="large">2 + <span class="hilite">15</span></td>
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<td class="large">=</td>
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<td class="large">
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<div class="center"><b>17</b></div>
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</td>
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<td class="large"> </td>
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</tr>
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<tr>
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<td class="large"><img src="images/style/no.svg" alt="not" height="30" width="30"></td>
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<td class="large"> </td>
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<td class="large">2 + 5 × 3 </td>
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<td class="large">=</td>
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<td class="large"><span class="wrong">7</span> × 3</td>
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<td class="large">=</td>
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<td class="large">
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<div class="center">21</div>
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</td>
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<td class="large">(wrong)</td>
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</tr>
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</tbody></table><br>
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<p class="large">Otherwise just go left to right</p>
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<table align="center" cellpadding="3" border="0">
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<tbody>
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<tr>
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<td class="large"><img src="images/style/yes.svg" alt="yes" height="30" width="30"></td>
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<td class="large"> </td>
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<td class="large"><span class="hilite">30 ÷ 5</span> × 3 </td>
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<td class="large">=</td>
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<td class="large"><span class="hilite">6</span> × 3</td>
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<td class="large">=</td>
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<td class="large">
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<div class="center"><b>18</b></div>
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</td>
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<td class="large"> </td>
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</tr>
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<tr>
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<td class="large"><img src="images/style/no.svg" alt="not" height="30" width="30"></td>
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<td class="large"> </td>
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<td class="large">30 ÷ 5 × 3 </td>
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<td class="large">=</td>
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<td class="large">30 ÷ <span class="wrong">15</span></td>
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<td class="large">=</td>
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<td class="large">
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<div class="center">2</div>
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</td>
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<td class="large">(wrong)</td>
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</tr>
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</tbody></table><br>
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<h2>How Do I Remember It All ... ? GEMS !</h2>
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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 class="large" width="60">
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<div class="center"><b>G</b></div>
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</td>
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<td><b>G</b>rouping () [] first</td>
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</tr>
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<tr>
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<td class="large">
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<div class="center"><b>E</b></div>
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</td>
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<td><b>E</b>xponents (ie Powers and Square Roots, etc.)</td>
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</tr>
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<tr>
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<td class="large">
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<div class="center"><b>M</b></div>
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</td>
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<td><b>M</b>ultiplication and Division (left-to-right)</td>
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</tr>
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<tr>
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<td class="large">
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<div class="center"><b>S</b></div>
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</td>
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<td>Addition and <b>S</b>ubtraction (left-to-right)</td>
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</tr>
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</tbody></table><br>
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<div class="center80">
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<p class="center"><span class="large">Divide and Multiply rank equally (and go left to right). </span></p>
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<p class="center"><span class="large">Add and Subtract rank equally (and go left to right) </span></p>
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</div>
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<p>So do it this way:</p>
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<div class="def">
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<p style="float:left; margin: 15px 30px 5px 5px;"><img src="algebra/images/gems.svg" alt="pemdas" height="106" width="155"></p>
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<p>After you have done "G" and "E", just go from left to right doing any "M" <i><b>or</b></i> "D" as you find them.</p>
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<p>Then go from left to right doing any "A" <i><b>or</b></i> "S" as you find them.</p><br>
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</div>
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<h2>Examples</h2>
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<div class="example">
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<h3>Example: How do you work out <b>3 + 6 × 2</b> ?</h3>
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<p><b>M</b>ultiplication before Addition:</p>
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<p>First <b>6 × 2 = 12</b>, then <b>3 + 12 = 15</b></p>
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</div><br>
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<div class="example">
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<h3>Example: How do you work out <b>(3 + 6) × 2</b> ?</h3>
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<p><b>G</b>rouping first:</p>
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<p>First <b>(3 + 6) = 9</b>, then <b>9 × 2 = 18</b></p>
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</div><br>
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<div class="example">
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<h3>Example: How do you work out <b>12 / 6 × 3 / 2</b> ?</h3>
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<p>Multiplication and Division rank equally, so just go left to right:</p>
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<p>First <b>12 / 6 = 2</b>, then <b>2 × 3 = 6</b>, then <b>6 / 2 = 3</b></p>
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</div>
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<p>A practical example:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="algebra/images/ball-throw.jpg" alt="ball throw" height="283" width="110"></p>
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<h3>Example: Sam threw a ball straight up at 20 meters per second, how far did it go in 2 seconds?</h3>
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<p>Sam uses this special formula that includes the effects of gravity:</p>
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<p class="center large">height = velocity × time − (1/2) × 9.8 × time<sup>2</sup></p>
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<p>Sam puts in the velocity of 20 meters per second and time of 2 seconds:</p>
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<p class="center large">height = 20 × 2 − (1/2) × 9.8 × 2<sup>2</sup></p>
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<p>Now for the calculations!</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">20 × 2 − (1/2) × 9.8 × 2<sup>2</sup></span></div>
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<div class="row"><span class="left"> <b>G</b>rouping first:</span><span class="right">20 × 2 − <span class="hilite">0.5</span> × 9.8 × 2<sup>2</sup></span></div>
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<div class="row"><span class="left">Then <b>E</b>xponents (2<sup>2</sup>=4):</span><span class="right">20 × 2 − 0.5 × 9.8 × <span class="hilite">4</span></span></div>
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<div class="row"><span class="left">Then the <b>M</b>ultiplies:</span><span class="right"><span class="hilite">40</span> − <span class="hilite">19.6</span></span></div>
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<div class="row"><span class="left"><b>S</b>ubtract and DONE !</span><span class="right"><b><span class="hilite">20.4</span></b></span></div>
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</div>
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<p><b>The ball reaches 20.4 meters after 2 seconds</b></p>
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</div>
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<h2>Exponents of Exponents ...</h2>
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<p>What about this example?</p>
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<p class="largest center">4<sup>3<sup>2</sup></sup></p>
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<p>Exponents are special: <b>they go top-down</b> (do the exponent at the top first). So we calculate this way:</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 style="text-align:right;">Start with:</td>
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<td> </td>
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<td style="text-align:center;"><span class="large">4<sup>3<sup>2</sup></sup></span></td>
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</tr>
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<tr>
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<td style="text-align:right;">3<sup>2</sup> = 3×3:</td>
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<td> </td>
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<td style="text-align:center;"><span class="large">4<sup>9</sup></span></td>
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</tr>
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<tr>
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<td style="text-align:right;">4<sup>9</sup> = 4×4×4×4×4×4×4×4×4:</td>
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<td> </td>
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<td class="large" align="center">262144</td>
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</tr>
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</tbody></table>
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<p>So 4<sup>3<sup>2</sup></sup> = 4<sup>(3<sup>2</sup>)</sup>, not (4<sup>3</sup>)<sup><sup>2</sup></sup></p>
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<p> </p>
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<p> </p>
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<p><i><b>And finally, what about the example from the beginning?</b></i></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">7 + (6 × 5<sup>2</sup> + 3)</span></div>
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<div class="row"><span class="left"><i>Grouping</i> first, inside that is <i>Exponents</i>:</span><span class="right">7 + (6 × <span class="hilite">25</span> + 3)</span></div>
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<div class="row"><span class="left">Then <i>Multiply</i>:</span><span class="right">7 + (<span class="hilite">150</span> + 3)</span></div>
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<div class="row"><span class="left">Then <i>Add</i>:</span><span class="right">7 + (<span class="hilite">153</span>)</span></div>
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<div class="row"><span class="left"><i><i>Grouping</i> </i>completed: </span><span class="right">7 + 153</span></div>
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<div class="row"><span class="left">Last operation is an <i>Add</i>:</span><span class="right"><b><span class="hilite">160</span></b></span></div>
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</div>
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<p> </p>
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<div class="questions">297, 1566, 1572, 298, 1568, 381, 1570, 3854, 3855, 383</div>
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<div class="related">
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