Question

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

Answer 1

$x-\frac{1}{x}=5$ .........(i)
$\Rightarrow {(x-\frac{1}{x})}^3=(5{)}^3$ [Cubing both sides of (i)]
$\Rightarrow x^3-\frac{1}{x^3}-3x.\frac{1}{x}.(x-\frac{1}{x})=125$
$\Rightarrow x^3-\frac{1}{x^3}-3(x-\frac{1}{x})=125$
$\Rightarrow x^3-\frac{1}{x^3}-3\times 5=125$ [Substituting, $(x-\frac{1}{x})=5]$
$\Rightarrow x^3-\frac{1}{x^3}-15=125$
$\Rightarrow x^3-\frac{1}{x^3}=(125+15)=140$