Question

If p = x ${}^{1/3}$ + x ${}^{-1/3}$ , then p${}^3$ - 3p is equal to

• A. 3
• B. (x + x^-1)/2
• C. x + x^-1
• D. -2
• E. 2(x + x^-1)

Answer 1

The correct option is C

x + x^-1

$\begin{array}{c}p=x^{1/3}+x^{-1/3}\\ p^3-3p={\left(x^{1/3}\right)}^3+{\left(x^{-1/3}\right)}^3+3x^{1/3}.x^{-1/3}\left(x^{1/3}+x^{1/3}\right)-3\left(x^{1/3}+x^{-1/3}\right)\\ ={\left(a+b\right)}^3=a^3+b^3+3ab\left(a+b\right)\\ =x+x^{-1}+3\left(x^{1/3}+x^{-1/3}\right)-3\left(x^{1/3}+x^{-1/3}\right)\\ =x+x^{-1}.\end{array}$