Question

If $x=3^{\frac{1}{3}}+3^{-\frac{1}{3}}$, then $3x^3-10$ is equal to:

• A. $-3x$
• B. $3x$
• C. $-9x$
• D. $9x$

Answer 1

The correct option is D $9x$

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

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

$\Rightarrow {(x-3^{\frac{1}{3}})}^3={\left(3^{-\frac{1}{3}}\right)}^3$

$\Rightarrow x^3-3.x.3^{\frac{1}{3}}(x-3^{-\frac{1}{3}})-{\left(3^{\frac{1}{3}}\right)}^3=3^{-1}$

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

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

$\Rightarrow x^3-3x=\frac{10}{3}$

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

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