https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10B_Problems/Problem_9&feed=atom&action=history
2003 AMC 10B Problems/Problem 9 - Revision history
2024-03-28T18:04:33Z
Revision history for this page on the wiki
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https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10B_Problems/Problem_9&diff=54463&oldid=prev
Nathan wailes at 16:09, 4 July 2013
2013-07-04T16:09:51Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 16:09, 4 July 2013</td>
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Nathan wailes
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10B_Problems/Problem_9&diff=43447&oldid=prev
Gina at 23:01, 26 November 2011
2011-11-26T23:01:22Z
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>-4x&=-12\\</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>-4x&=-12\\</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>x&=\boxed{\<del class="diffchange diffchange-inline">mathrm</del>{(B) \ } 3}\end{align*}</cmath></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>x&=\boxed{\<ins class="diffchange diffchange-inline">textbf</ins>{(B) \ } 3}\end{align*}</cmath></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{AMC10 box|year=2003|ab=B|num-b=8|num-a=10}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{AMC10 box|year=2003|ab=B|num-b=8|num-a=10}}</div></td></tr>
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Gina
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10B_Problems/Problem_9&diff=39449&oldid=prev
Gina at 07:35, 10 June 2011
2011-06-10T07:35:36Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 07:35, 10 June 2011</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Manipulate the powers of <math>5</math> in order to get a clean expression.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Manipulate the powers of <math>5</math> in order to get a clean expression.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><<del class="diffchange diffchange-inline">math</del>><del class="diffchange diffchange-inline">25^{-2} = </del>\frac{5^{48<del class="diffchange diffchange-inline">/</del>x}}{5^{26<del class="diffchange diffchange-inline">/</del>x} \cdot 25^{17<del class="diffchange diffchange-inline">/</del>x}}<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins class="diffchange diffchange-inline">cmath</ins>>\frac{5^<ins class="diffchange diffchange-inline">{\frac</ins>{48<ins class="diffchange diffchange-inline">}{</ins>x<ins class="diffchange diffchange-inline">}</ins>}}{5^<ins class="diffchange diffchange-inline">{\frac</ins>{26<ins class="diffchange diffchange-inline">}{</ins>x<ins class="diffchange diffchange-inline">}</ins>} \cdot 25^<ins class="diffchange diffchange-inline">{\frac</ins>{17<ins class="diffchange diffchange-inline">}{</ins>x}}} = \frac{5^<ins class="diffchange diffchange-inline">{\frac</ins>{48<ins class="diffchange diffchange-inline">}{</ins>x<ins class="diffchange diffchange-inline">}</ins>}}{5^<ins class="diffchange diffchange-inline">{\frac</ins>{26<ins class="diffchange diffchange-inline">}{</ins>x<ins class="diffchange diffchange-inline">}</ins>} \cdot 5^{<ins class="diffchange diffchange-inline">\frac{34}{</ins>x}}<ins class="diffchange diffchange-inline">} = </ins>5^{\frac{<ins class="diffchange diffchange-inline">48}</ins>{x}<ins class="diffchange diffchange-inline">-(\frac{26</ins>}{x}<ins class="diffchange diffchange-inline">+</ins>\<ins class="diffchange diffchange-inline">frac</ins>{34<ins class="diffchange diffchange-inline">}{</ins>x}<ins class="diffchange diffchange-inline">)</ins>} <ins class="diffchange diffchange-inline">= </ins>5^{-<ins class="diffchange diffchange-inline">\frac{12</ins>}{x<ins class="diffchange diffchange-inline">}</ins>}</<ins class="diffchange diffchange-inline">cmath</ins>></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins class="diffchange diffchange-inline">cmath</ins>><ins class="diffchange diffchange-inline">25^{-2} = (</ins>5<ins class="diffchange diffchange-inline">^2)</ins>^{-<ins class="diffchange diffchange-inline">2</ins>} = 5^{-<ins class="diffchange diffchange-inline">4</ins>}</<ins class="diffchange diffchange-inline">cmath</ins>></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><math>(5^2)^{-2</del>} = \frac{5^{48<del class="diffchange diffchange-inline">/</del>x}}{5^{26<del class="diffchange diffchange-inline">/</del>x} \cdot <del class="diffchange diffchange-inline">(</del>5^<del class="diffchange diffchange-inline">2)^</del>{<del class="diffchange diffchange-inline">17/</del>x}}<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins class="diffchange diffchange-inline">cmath</ins>>5^{-4} = 5^{-<ins class="diffchange diffchange-inline">\frac{</ins>12<ins class="diffchange diffchange-inline">}{</ins>x<ins class="diffchange diffchange-inline">}</ins>}</<ins class="diffchange diffchange-inline">cmath</ins>></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><math></del>5^{<del class="diffchange diffchange-inline">-4} = </del>\frac{<del class="diffchange diffchange-inline">5^</del>{<del class="diffchange diffchange-inline">48/</del>x}}{<del class="diffchange diffchange-inline">5^{26/</del>x} \<del class="diffchange diffchange-inline">cdot 5^</del>{34<del class="diffchange diffchange-inline">/</del>x}}<del class="diffchange diffchange-inline"></math></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><math></del>5^{-<del class="diffchange diffchange-inline">4</del>} <del class="diffchange diffchange-inline">= 5^</del>{<del class="diffchange diffchange-inline">48/x-26/x-34/</del>x}</<del class="diffchange diffchange-inline">math</del>></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><<del class="diffchange diffchange-inline">math</del>>5^{-<del class="diffchange diffchange-inline">4</del>} = 5^{<del class="diffchange diffchange-inline">(48</del>-<del class="diffchange diffchange-inline">26-34)/x</del>}</<del class="diffchange diffchange-inline">math</del>></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><<del class="diffchange diffchange-inline">math</del>>5^{-4} = 5^{-12<del class="diffchange diffchange-inline">/</del>x}</<del class="diffchange diffchange-inline">math</del>></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If two numbers are equal, and their bases are equal, then their exponents are equal as well.  Set the two exponents equal to each other.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If two numbers are equal, and their bases are equal, then their exponents are equal as well.  Set the two exponents equal to each other.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><<del class="diffchange diffchange-inline">math</del>>-4=\frac{-12}{x}<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins class="diffchange diffchange-inline">cmath</ins>><ins class="diffchange diffchange-inline">\begin{align*}</ins>-4<ins class="diffchange diffchange-inline">&</ins>=\frac{-12}{x}<ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>-4x<ins class="diffchange diffchange-inline">&</ins>=-12<ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><math></del>-4x=-12<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">x&=</ins>\boxed{\<ins class="diffchange diffchange-inline">mathrm</ins>{(B) <ins class="diffchange diffchange-inline">\ </ins>} <ins class="diffchange diffchange-inline">3}\end{align*</ins>}</<ins class="diffchange diffchange-inline">cmath</ins>></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><math></del>\boxed{<del class="diffchange diffchange-inline">x=3 </del>\<del class="diffchange diffchange-inline">text</del>{ (B)}}</<del class="diffchange diffchange-inline">math</del>></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{AMC10 box|year=2003|ab=B|num-b=8|num-a=10}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{AMC10 box|year=2003|ab=B|num-b=8|num-a=10}}</div></td></tr>
</table>
Gina
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10B_Problems/Problem_9&diff=39167&oldid=prev
Djmathman: Created page with "==Problem== Find the value of <math>x</math> that satisfies the equation <math>25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}.</math> <math>\textbf{(A) } 2 \qquad\textbf{(..."
2011-06-04T21:10:30Z
<p>Created page with "==Problem== Find the value of <math>x</math> that satisfies the equation <math>25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}.</math> <math>\textbf{(A) } 2 \qquad\textbf{(..."</p>
<p><b>New page</b></p><div>==Problem==<br />
<br />
Find the value of <math>x</math> that satisfies the equation<br />
<math>25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}.</math><br />
<br />
<math>\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 9 </math><br />
<br />
==Solution==<br />
Manipulate the powers of <math>5</math> in order to get a clean expression.<br />
<br />
<math>25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}</math><br />
<br />
<math>(5^2)^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot (5^2)^{17/x}}</math><br />
<br />
<math>5^{-4} = \frac{5^{48/x}}{5^{26/x} \cdot 5^{34/x}}</math><br />
<br />
<math>5^{-4} = 5^{48/x-26/x-34/x}</math><br />
<br />
<math>5^{-4} = 5^{(48-26-34)/x}</math><br />
<br />
<math>5^{-4} = 5^{-12/x}</math><br />
<br />
If two numbers are equal, and their bases are equal, then their exponents are equal as well. Set the two exponents equal to each other.<br />
<br />
<math>-4=\frac{-12}{x}</math><br />
<br />
<math>-4x=-12</math><br />
<br />
<math>\boxed{x=3 \text{ (B)}}</math><br />
<br />
==See Also==<br />
{{AMC10 box|year=2003|ab=B|num-b=8|num-a=10}}</div>
Djmathman