solve
X to the power 15 + 1/ X to the power 25
when X + 1/X = 1
So you want to find the value of $\displaystyle x^{15} + \frac{1}{x^{25}}$ given that $\displaystyle x + \frac{1}{x} = 1$.
A brutal approach:
$\displaystyle x + \frac{1}{x} = 1 \Rightarrow x^2 - x + 1 = 0 \Rightarrow x = ......$. Express the solution in polar form.
$\displaystyle x^{15} + \frac{1}{x^{25}} = \frac{1}{x^{25}} (x^{40} + 1)$. Substitute the polar form of x, use de Moivre's theorem and tidy up.