Difference between revisions of "AoPS Wiki talk:Problem of the Day/September 7, 2011"
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<cmath>-1(-2)=\left(x^4+\frac{1}{x^4}\right)\left(x^3+\frac{1}{x^3}\right)=x^7+\frac{1}{x^7}+x+\frac{1}{x}=x^7+\frac{1}{x^7}+1</cmath> | <cmath>-1(-2)=\left(x^4+\frac{1}{x^4}\right)\left(x^3+\frac{1}{x^3}\right)=x^7+\frac{1}{x^7}+x+\frac{1}{x}=x^7+\frac{1}{x^7}+1</cmath> | ||
and therefore, <math>x^7+\frac{1}{x^7}=2-1=\boxed{1}</math>. | and therefore, <math>x^7+\frac{1}{x^7}=2-1=\boxed{1}</math>. | ||
+ | <noinclude>[[category:Problem of the Day]]</noinclude> |
Revision as of 08:24, 7 September 2011
Solution
Raising that thing to the seventh doesn't look appealing, nor does solving for . So we try getting to
gradually.
Since ,
and thus
.
We can also compute:
so
.
Squaring , we see
and thus
. Thus,
and therefore,
.