Question

Rationalize the denominator and simplify$\frac{\sqrt{10}}{\sqrt{5}-2}$

Short answer

$5\sqrt{2}+2\sqrt{10}$

Step-by-step solution

Step 1. Concept of rationalize.

Let be a positive number then $\frac{1}{\sqrt{a}}$can be rationalize in the denominator as$\frac{1}{\sqrt{a}}\frac{\sqrt{a}}{\sqrt{a}}=\frac{\sqrt{a}}{a}$

Step 2. Given expression.

The expression is$\frac{\sqrt{10}}{\sqrt{5}-2}$

Step 3. Solving the expression.

$\begin{array}{l}\frac{\sqrt{10}}{\sqrt{5}-2}\\ \frac{\sqrt{10}(\sqrt{5}+2)}{(\sqrt{5}-2)(\sqrt{5}+2)}(\mathrm{Rationalize}\mathrm{the}\mathrm{denominator})\\ \frac{\sqrt{10}(\sqrt{5}+2)}{5-4}\left(\mathrm{Simplify}\right)\\ \sqrt{10}\sqrt{5}+2\sqrt{10}\\ =5\sqrt{2}+2\sqrt{10}\end{array}$