Rationalize the denominator and simplify.

Question

Quizplus · Accepted Answer

To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of the denominator is $\sqrt[3]{9+x}-\sqrt[3]{9-x}$.

\begin{align*}
\frac{1}{\sqrt[3]{(9-x)^2} + \sqrt[3]{(9+x)^2} + \sqrt[3]{(81-x^2)}} &= \frac{1}{(\sqrt[3]{9-x}-\sqrt[3]{9+x})(\sqrt[3]{(9-x)^2} + \sqrt[3]{(9+x)^2} + \sqrt[3]{(81-x^2)})} \
&= \frac{1}{(9-x) + (9+x) + \sqrt[3]{(9-x)(9+x)^2} - (9+x) - (9-x) + \sqrt[3]{(9-x)^2(9+x)} + \sqrt[3]{(81-x^2)}} \
&= \frac{1}{2\sqrt[3]{81-x^2}}
\end{align*}

Therefore, the answer is (C).

Rationalize the Denominator and Simplify