Question

If $p=2^{\frac{2}{3}}+2^{\frac{1}{3}}$,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

The correct option is C $p^3-6p-6=0$

Given,

$p=2^{\frac{2}{3}}+2^{\frac{1}{3}}$

cubing both sides

$p^3=(2^{\frac{2}{3}}+2^{\frac{1}{3}}{)}^3$

$p^3={\left(2^{\frac{2}{3}}\right)}^3+3{\left(2^{\frac{2}{3}}\right)}^2\cdot 2^{\frac{1}{3}}+3\cdot 2^{\frac{2}{3}}{\left(2^{\frac{1}{3}}\right)}^2+{\left(2^{\frac{1}{3}}\right)}^3$

$p^3=6+6\cdot 2^{\frac{2}{3}}+6\cdot 2^{\frac{1}{3}}$

$p^3=6+6(2^{\frac{2}{3}}+2^{\frac{1}{3}})$

$p^3=6+6p$

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