<s>[INST] finding $x^6+y^6$ given $x+y$ and $xy$ Here is the question
If $x + y = 4$ and $xy = 2$, then find $x^6+ y^6$.
This is from a previous timed competition, so fastest answers are the best answers.
I've tried using sum of cubes, but I dont know what to do after $(x^2+y^2)(x^4-x^2y^2+y^4)$
. The only other way I can think of is solving for x and y, but that wouldn't be too quick. Any help?
 [/INST] Your sum is$$\begin{align}\sum_\pm(2\pm\sqrt{2})^6&=2(2^6+\sqrt{2}^6+15(2^4\sqrt{2}^2+2^2\sqrt{2}^4))\\&=2(64+8+15(2\sqrt{2})^2(2^2+2))\\&=2(72+15\cdot8\cdot6)\\&=1584.\end{align}$$
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