Question

In the following exercises, simplify.

$\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}$

Short answer

The answer is$\frac{{\left(\sqrt{x}+2\sqrt{2}\right)}^2}{x-8}$

Step-by-step solution

Step 1. Given

The expression is$\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}$

Step 2. Rationalize the denominators of the given expressions.

$$\begin{gathered} \frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}} \\ =\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}\times \frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}+\sqrt{8}} \\ =\frac{{\left(\sqrt{x}+\sqrt{8}\right)}^2}{{\left(\sqrt{x}\right)}^2-{\left(\sqrt{8}\right)}^2} \\ =\frac{{\left(\sqrt{x}+2\sqrt{2}\right)}^2}{x-8} \end{gathered}$$