Question

If $x-\frac{1}{x}=7,$ find the value of $x^3-\frac{1}{x^3}$.

Answer 1

$x-\frac{1}{x}=7$

Cubing both sides,

${(x-\frac{1}{x})}^3=(7{)}^3\Rightarrow x^3-\frac{1}{x^3}-3(x-\frac{1}{x})=343\Rightarrow x^3-\frac{1}{x^3}-3\times 7=343\Rightarrow x^3-\frac{1}{x^3}-21=343\Rightarrow x^3-\frac{1}{x^3}=343+21=364$