Question

Choose the correct answer from the alternatives given.
If $\frac{a}{b}-\frac{b}{a}$ = 3, then find the value of $\frac{a^3}{b^3}-\frac{b^3}{a^3}$

• A. 10$\sqrt{3}$
• B. 10$\sqrt{13}$
• C. 13$\sqrt{10}$
• D. 13$\sqrt{3}$

Answer 1

Answer: (B)

10$\sqrt{13}$

$\frac{a}{b}-\frac{b}{a}=3$ ----- ( 1 ) [ Given ]

$\Rightarrow$ ${(\frac{a}{b}-\frac{b}{a})}^3=(3{)}^3$

$\Rightarrow$ $\frac{a^3}{b^3}-\frac{b^3}{a^3}-3\times \frac{a}{b}\times \frac{b}{a}(\frac{a}{b}-\frac{b}{a})=27$

$\Rightarrow$ $\frac{a^3}{b^3}-\frac{b^3}{a^3}-3\times (\frac{a}{b}-\frac{b}{a})=27$

$\Rightarrow$ $\frac{a^3}{b^3}-\frac{b^3}{a^3}-3\times \left(3\right)=27$

$\Rightarrow$ $\frac{a^3}{b^3}-\frac{b^3}{a^3}-9=27$

$\Rightarrow$ $\frac{a^3}{b^3}-\frac{b^3}{a^3}=36$

$\Rightarrow$ $\frac{a^3}{b^3}-\frac{b^3}{a^3}=10\sqrt{13}$