https://artofproblemsolving.com/wiki/index.php?title=1990_AHSME_Problems/Problem_22&feed=atom&action=history
1990 AHSME Problems/Problem 22 - Revision history
2024-03-29T07:32:00Z
Revision history for this page on the wiki
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Zimbalono at 17:17, 4 February 2016
2016-02-04T17:17:10Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:17, 4 February 2016</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>== Solution ==</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>== Solution ==</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><math>\fbox{D}</math></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">This equation is <math>r^6e^{6\theta i}=2^6e^{(\pi\pm 2k\pi) i}</math>. Solving in the usual way, <math>r=2</math> and <math>\theta\in\{\pm 30^\circ,\pm 90^\circ,\pm 150^\circ\}</math>.</ins></div></td></tr>
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<tr><td colspan="2"> </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">Thus there are only two solutions with positive real part, and they are conjugates, so their product is <math>r^2=4</math> which is </ins><math>\fbox{D}</math></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>
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Zimbalono
https://artofproblemsolving.com/wiki/index.php?title=1990_AHSME_Problems/Problem_22&diff=63888&oldid=prev
Timneh: Created page with "== Problem == If the six solutions of <math>x^6=-64</math> are written in the form <math>a+bi</math>, where <math>a</math> and <math>b</math> are real, then the product of those..."
2014-09-29T19:46:58Z
<p>Created page with "== Problem == If the six solutions of <math>x^6=-64</math> are written in the form <math>a+bi</math>, where <math>a</math> and <math>b</math> are real, then the product of those..."</p>
<p><b>New page</b></p><div>== Problem ==<br />
<br />
If the six solutions of <math>x^6=-64</math> are written in the form <math>a+bi</math>, where <math>a</math> and <math>b</math> are real, then the product of those solutions with <math>a>0</math> is<br />
<br />
<br />
<math>\text{(A) } -2\quad<br />
\text{(B) } 0\quad<br />
\text{(C) } 2i\quad<br />
\text{(D) } 4\quad<br />
\text{(E) } 16</math><br />
<br />
== Solution ==<br />
<math>\fbox{D}</math><br />
<br />
== See also ==<br />
{{AHSME box|year=1990|num-b=21|num-a=23}} <br />
<br />
[[Category: Intermediate Algebra Problems]]<br />
{{MAA Notice}}</div>
Timneh