Question

If $x=3^{\frac{1}{3}}+3^{\frac{-1}{3}}$ then find the value of $3x^3-9x$

• A. $3$
• B. $9$
• C. $12$
• D. $10$

Answer 1

The correct option is D $10$

We have $x=3^{\frac{1}{3}}+3^{\frac{-1}{3}}$

Cubing both sides, we get

$x^3={(3^{\frac{1}{3}}+3^{\frac{-1}{3}})}^3$

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

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

$\Rightarrow 3x^3=9+1+9x$

$\Rightarrow 3x^3-9x=10$