<s>[INST] If $x+\frac1x=5$ find $x^5+\frac1{x^5}$. If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$.
Is there any other way find it?
$$
\left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110.
$$
Thanks
 [/INST] Hint: Expand (binomial formula) $\left(x+\frac{1}{x}\right)^3$ and then $\left(x+\frac{1}{x}\right)^5$. (No need for even powers.)
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