<s>[INST] Find $x^4+y^4$ and $x^3+y^3$ if $x+y=2$ and $x^2+y^2=8$ Find $x^4+y^4$ if $x+y=2$ and $x^2+y^2=8$
So i started the problem by nothing that $x^2+y^2=(x+y)^2 - 2xy$ but that doesn't help!
I also seen that $x+y=2^1$ and $x^2+y^2=2^3$ so maybe $x^3+y^3=2^5$ and $x^4+y^4=2^7$ but i think this is just coincidence
So how can i solve this problem?
PLEASE i need some help and thanks for all!!
 [/INST] \begin{eqnarray}
x^4+y^4&=&(x^2+y^2)^2-2x^2y^2=(x^2+y^2)^2-\frac12(2xy)^2=(x^2+y^2)^2-\frac12\left[(x+y)^2-(x^2+y^2)\right]^2\\
&=&8^2-\frac12(2^2-8)^2=56.
\end{eqnarray}
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