# How do you rationalize the denominator and simplify 5 / ( sqrt 14 - 2 )?

Apr 8, 2016

$\frac{\sqrt{14} + 2}{2} = \frac{\sqrt{14}}{2} + 1$

#### Explanation:

By rationalizing a fraction, we mean removing any irrational values from the denominator, without changing the fraction.

Here, we have to remove $\sqrt{14}$ from the denominator, without changing the value of the expression.

We know that ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

We have $\sqrt{14} - 2$ as the denominator. To remove the square root, we must multiply the denominator by $\sqrt{14} + 2$

Dividing and multiplying a fraction by the same number does not change the fraction.

$\frac{5}{\sqrt{14} - 2} \times \frac{\sqrt{14} + 2}{\sqrt{14} + 2}$

$\frac{5 \times \left(\sqrt{14} + 2\right)}{\left(\sqrt{14} - 2\right) \left(\sqrt{14} + 2\right)}$

$\frac{5 \sqrt{14} + 10}{{\left(\sqrt{14}\right)}^{2} - {2}^{2}}$

$\frac{5 \sqrt{14} + 10}{14 - 4} = \frac{5 \sqrt{14} + 10}{10}$

5 is the common factor of the numerator.

$\frac{5 \left(\sqrt{14} + 2\right)}{10}$

Cancel 5 from the numerator and denominator.

$\frac{\sqrt{14} + 2}{2} = \frac{\sqrt{14}}{2} + 1$