Question

If $p=2^{2/3}+2^{1/3}$, then:

• A. $p^3-6p+6=0$
• B. $p^3-3p-6=0$
• C. $p^3-6p-6=0$
• D. $p^3-3p+6=0$

Answer 1

Answer: (C)

$p^3-6p-6=0$

We know, $(x+y{)}^3=x^3+y^3+3xy(x+y)$.

Given, $p=2^{2/3}+2^{1/3}$.

On taking cube both sides, we get,

$p^3=(2^{2/3}+2^{1/3}{)}^3$
$\Rightarrow p^3=(2^{2/3}{)}^3+(2^{1/3}{)}^3+3\left(2^{2/3}\right)\left(2^{1/3}\right)(2^{2/3}+2^{1/3})$
$\Rightarrow p^3=4+2+3\left(2\right)\left(p\right)$

$\Rightarrow p^3=6+6p$
$\Rightarrow p^3-6p-6=0$.

Therefore, option $C$ is correct.