# How do you rationalize the denominator and simplify (3-sqrt5)/(4+sqrt5)?

Mar 17, 2016

$\frac{17 - 7 \sqrt{5}}{11}$

#### Explanation:

$1$. Start by multiplying the numerator and denominator by the conjugate of the fraction's denominator, $4 - \sqrt{5}$.

$\frac{3 - \sqrt{5}}{4 + \sqrt{5}}$

$= \frac{3 - \sqrt{5}}{4 + \sqrt{5}} \left(\frac{4 - \sqrt{5}}{4 - \sqrt{5}}\right)$

$2$. Simplify the numerator.

$= \frac{12 - 3 \sqrt{5} - 4 \sqrt{5} + 5}{4 + \sqrt{5}} \left(\frac{1}{4 - \sqrt{5}}\right)$

$= \frac{17 - 7 \sqrt{5}}{4 + \sqrt{5}} \left(\frac{1}{4 - \sqrt{5}}\right)$

$3$. Simplify the denominator. Note that it contains a difference of squares $\left(\textcolor{red}{{a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)}\right)$.

$= \frac{17 - 7 \sqrt{5}}{16 - 5}$

$= \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \frac{17 - 7 \sqrt{5}}{11} \textcolor{w h i t e}{\frac{a}{a}} |}}}$