Difference between revisions of "1990 AHSME Problems/Problem 22"
(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...") |
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== Solution == | == Solution == | ||
| − | <math>\fbox{D}</math> | + | 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>. |
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| + | 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 <math>\fbox{D}</math> | ||
== See also == | == See also == | ||
Latest revision as of 13:17, 4 February 2016
Problem
If the six solutions of 
are written in the form 
, where 
and 
are real, then the product of those solutions with 
is
Solution
This equation is 
. Solving in the usual way, 
and 
.
Thus there are only two solutions with positive real part, and they are conjugates, so their product is 
which is 
See also
| 1990 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
