Simplify by rationalizing the denominator.
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Question

Quizplus · Accepted Answer

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{5} - \sqrt[3]{7}$. This gives us:
$$\frac{\sqrt{5}}{5 - \sqrt[3]{343}} \cdot \frac{\sqrt{5} + \sqrt[3]{7}}{\sqrt{5} + \sqrt[3]{7}} = \frac{\sqrt{5}(\sqrt{5}+\sqrt[3]{7})}{5(\sqrt{5}+\sqrt[3]{7})-(\sqrt[3]{343})(\sqrt{5}+\sqrt[3]{7})}$$
Simplifying the denominator:
\begin{align*}
5(\sqrt{5}+\sqrt[3]{7})-(\sqrt[3]{343})(\sqrt{5}+\sqrt[3]{7}) &= 5\sqrt{5}+5\sqrt[3]{7}-\sqrt[3]{7}\sqrt{5}-\sqrt[3]{7}\sqrt[3]{343} \
&= 5\sqrt{5}+5\sqrt[3]{7}-\sqrt[3]{7}\sqrt{5}-7 \
&= (5-\sqrt[3]{7})\sqrt{5}+5\sqrt[3]{7}-7
\end{align*}
So the simplified fraction is:
$$\frac{\sqrt{5}(\sqrt{5}+\sqrt[3]{7})}{(5-\sqrt[3]{7})\sqrt{5}+5\sqrt[3]{7}-7}$$
which matches choice C.

Simplify by Rationalizing the Denominator