Question

If $a-\frac{1}{a}=3\sqrt{2}$, If find the value of $a^3-\frac{1}{a^3}$.

Answer 1

$a-\frac{1}{a}=3\sqrt{2}----\left(1\right)$

To find : $a^3-\frac{1}{a^3}$

cubing both sides of eq $\left(1\right)$

${(a-\frac{1}{a})}^3=(3\sqrt{2}{)}^3$ $\left[\right(a-b{)}^3=a^3-b^3-3ab(a-b)]$

$a^3-\frac{1}{a^3}-\frac{3a}{a}[a-\frac{1}{a}]=54\sqrt{2}$

$a^3-\frac{1}{a^3}-9\sqrt{2}=54\sqrt{2}$

$a^3-\frac{1}{a^3}=63\sqrt{2}$