# How do you rationalize the denominator of (7-sqrt5)/(7+sqrt5)?

##### 1 Answer
Jan 22, 2016

$\frac{27 - 7 \sqrt{5}}{22}$

#### Explanation:

To rationalize the denominator, you should take advantage of the formula

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

To do so, you need to expand your fraction with $7 - \sqrt{5}$:

$\frac{7 - \sqrt{5}}{7 + \sqrt{5}} = \frac{\left(7 - \sqrt{5}\right) \textcolor{b l u e}{\left(7 - \sqrt{5}\right)}}{\left(7 + \sqrt{5}\right) \textcolor{b l u e}{\left(7 - \sqrt{5}\right)}} = {\left(7 - \sqrt{5}\right)}^{2} / \left(\left(7 + \sqrt{5}\right) \left(7 - \sqrt{5}\right)\right)$

To simplify the numerator, apply the formula

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

To simplify the denominator, apply the formula

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

Thus, you will get:

$\ldots = \frac{{7}^{2} - 2 \cdot 7 \cdot \sqrt{5} + {\left(\sqrt{5}\right)}^{2}}{{7}^{2} - {\left(\sqrt{5}\right)}^{2}} = \frac{49 - 14 \sqrt{5} + 5}{49 - 5}$

$= \frac{54 - 14 \sqrt{5}}{44} = \frac{2 \left(27 - 7 \sqrt{5}\right)}{2 \cdot 22} = \frac{\cancel{2} \left(27 - 7 \sqrt{5}\right)}{\cancel{2} \cdot 22}$

$= \frac{27 - 7 \sqrt{5}}{22}$