https://artofproblemsolving.com/wiki/index.php?title=1973_AHSME_Problems/Problem_13&feed=atom&action=history
1973 AHSME Problems/Problem 13 - Revision history
2024-03-29T05:20:48Z
Revision history for this page on the wiki
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Made in 2016: /* See Also */
2020-02-20T17:59:44Z
<p><span dir="auto"><span class="autocomment">See Also</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:59, 20 February 2020</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>==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="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>{{AHSME <del class="diffchange diffchange-inline">35p </del>box|year=1973|num-b=12|num-a=14}}</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>{{AHSME <ins class="diffchange diffchange-inline">30p </ins>box|year=1973|num-b=12|num-a=14}}</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>[[Category:Introductory Algebra Problems]]</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>[[Category:Introductory Algebra Problems]]</div></td></tr>
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Made in 2016
https://artofproblemsolving.com/wiki/index.php?title=1973_AHSME_Problems/Problem_13&diff=96535&oldid=prev
Rockmanex3: /* Solution */
2018-07-27T21:32:41Z
<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:32, 27 July 2018</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;"></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>Squaring the expression and taking the positive square root (since numerator and denominator are positive) of the result yields</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>Squaring the expression and taking the positive square root (since numerator and denominator are positive) of the result yields</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><cmath>\sqrt{(\frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}})^2}</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><cmath>\sqrt{<ins class="diffchange diffchange-inline">\left</ins>(\frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}}<ins class="diffchange diffchange-inline">\right</ins>)^2}</cmath></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><cmath>\sqrt{\frac{4(2+4\sqrt{3}+6)}{9(2+\sqrt{3}}}</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><cmath>\sqrt{\frac{4(2+4\sqrt{3}+6)}{9(2+\sqrt{3}<ins class="diffchange diffchange-inline">)</ins>}}</cmath></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><cmath>\sqrt{\frac{4(8+4\sqrt{3})}{9(2+\sqrt{3}}}</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><cmath>\sqrt{\frac{4(8+4\sqrt{3})}{9(2+\sqrt{3}<ins class="diffchange diffchange-inline">)</ins>}}</cmath></div></td></tr>
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Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1973_AHSME_Problems/Problem_13&diff=96534&oldid=prev
Rockmanex3: Solution to Problem 13 (credit to gaussintraining)
2018-07-27T21:31:19Z
<p>Solution to Problem 13 (credit to gaussintraining)</p>
<p><b>New page</b></p><div>==Problem==<br />
<br />
The fraction <math> \frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}} </math> is equal to <br />
<br />
<math> \textbf{(A)}\ \frac{2\sqrt2}{3} \qquad<br />
\textbf{(B)}\ 1 \qquad<br />
\textbf{(C)}\ \frac{2\sqrt3}3 \qquad<br />
\textbf{(D)}\ \frac43 \qquad<br />
\textbf{(E)}\ \frac{16}{9}</math><br />
<br />
==Solution==<br />
<br />
Squaring the expression and taking the positive square root (since numerator and denominator are positive) of the result yields<br />
<cmath>\sqrt{(\frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}})^2}</cmath><br />
<cmath>\sqrt{\frac{4(2+4\sqrt{3}+6)}{9(2+\sqrt{3}}}</cmath><br />
<cmath>\sqrt{\frac{4(8+4\sqrt{3})}{9(2+\sqrt{3}}}</cmath><br />
<cmath>\sqrt{\frac{16}{9}}</cmath><br />
<cmath>\frac43</cmath><br />
<br />
The answer is <math>\boxed{\textbf{(D)}}</math>.<br />
<br />
==See Also==<br />
{{AHSME 35p box|year=1973|num-b=12|num-a=14}}<br />
<br />
[[Category:Introductory Algebra Problems]]</div>
Rockmanex3