Perform the division by rationalizing the denominator and simplifying.

Question

Quizplus · Accepted Answer

We cannot directly divide by a square root in the denominator, so we must rationalize by multiplying both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of $\sqrt{10} - \sqrt{6}$ is $\sqrt{10} + \sqrt{6}$. 
\begin{align*}
\frac{1}{\sqrt{10}-\sqrt{6}} &= \frac{1}{\sqrt{10}-\sqrt{6}} \cdot \frac{\sqrt{10}+\sqrt{6}}{\sqrt{10}+\sqrt{6}} \
&= \frac{\sqrt{10}+\sqrt{6}}{10 - 6} \
&= \frac{\sqrt{10}+\sqrt{6}}{4}
\end{align*}
Therefore, the answer is option C.

Perform the Division by Rationalizing the Denominator and Simplifying