An algebra problem by A Former Brilliant Member

Given

$x+\dfrac{1}{x}=5$

Squaring both sides

$\left(x+\dfrac{1}{x}\right)^2=5^2$

$x^2+2+\dfrac{1}{x^2}=25$

$x^2+\dfrac{1}{x^2}=23$

Using the same process other two times

$x^4+\dfrac{1}{x^4}=23^2-2=527$

$x^8+\dfrac{1}{x^8}=527^2-2=\boxed{277727}$