Question

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

Answer 1

$a^3+b^3=(a+b)(a^2-ab+b^2)$

$\Rightarrow x^3+\frac{1}{x^3}=(x+\frac{1}{x})[x^2+\frac{1}{x^2}-1]$

$\Rightarrow (x^3+\frac{1}{x^3})=5[{(x+\frac{1}{x})}^2-2-1]$

$=5[25-3]$

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