AoPS Wiki talk:Problem of the Day/September 7, 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:
![\[1(-1)=\left(x+\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}\right)=x^3+\frac{1}{x^3}+x+\frac{1}{x}=x^3+\frac{1}{x^3}+1\]](http://latex.artofproblemsolving.com/c/3/d/c3d26fd0e81138a80aa75efd8bab712cc0667bbb.png)
so 
.
Squaring 
, we see
![\[1=\left(x^2+\frac{1}{x^2}\right)^2=x^4+\frac{1}{x^4}+2\]](http://latex.artofproblemsolving.com/e/1/1/e112358c4ae2b61cd11cbd79c2e4d84d6dfb8ee8.png)
and thus 
. Thus,
![\[-1(-2)=\left(x^4+\frac{1}{x^4}\right)\left(x^3+\frac{1}{x^3}\right)=x^7+\frac{1}{x^7}+x+\frac{1}{x}=x^7+\frac{1}{x^7}+1\]](http://latex.artofproblemsolving.com/e/7/7/e77e051196cb4a74b2990a3a770ee2199597245d.png)
and therefore, 
.
