Question

In the following exercises, simplify.

$\frac{\sqrt{2}}{\sqrt{x}-\sqrt{6}}$

Short answer

The answer is$\frac{\sqrt{2}\left(\sqrt{x}+\sqrt{6}\right)}{x-6}$

Step-by-step solution

Step 1. Given

The expression is$\frac{\sqrt{2}}{\sqrt{x}-\sqrt{6}}$

Step 2. Rationalize the denominators of the given expressions.

$$\begin{gathered} \frac{\sqrt{2}}{\sqrt{x}-\sqrt{6}} \\ =\frac{\sqrt{2}}{\sqrt{x}-\sqrt{6}}\times \frac{\sqrt{x}+\sqrt{6}}{\sqrt{x}+\sqrt{6}} \\ =\frac{\sqrt{2}\left(\sqrt{x}+\sqrt{6}\right)}{{\left(\sqrt{x}\right)}^2-{\left(\sqrt{6}\right)}^2} \\ =\frac{\sqrt{2}\left(\sqrt{x}+\sqrt{6}\right)}{x-6} \end{gathered}$$