Question

If$x+\frac{1}{x}=5$, find the value of $x^3+\frac{1}{x^3}$.

Answer 1

We have, $x+\frac{1}{x}=5$

on Cubing both sides, we get

${(x+\frac{1}{x})}^3=(5{)}^3$

$\Rightarrow x^3+\frac{1}{x^3}+3\times x\times \frac{1}{x}(x+\frac{1}{x})=125$ [$\because (a+b{)}^3=a^3+b^3+3ab(a+b)$]

$\Rightarrow x^3+\frac{1}{x^3}+3\times 5=125$

$\Rightarrow x^3+\frac{1}{x^3}+15=125$

$\Rightarrow x^3+\frac{1}{x^3}=125-15=110$