# How do you rationalize the denominator and simplify 2/(5-sqrt3)?

##### 1 Answer
Mar 28, 2016

$\frac{5 + \sqrt{3}}{11}$

#### Explanation:

$1$. Multiply the numerator and denominator by the conjugate of $5 - \sqrt{3}$, which is $5 + \sqrt{3}$.

$\frac{2}{5 - \sqrt{3}}$

$= \frac{2}{5 - \sqrt{3}} \left(\frac{5 + \sqrt{3}}{5 + \sqrt{3}}\right)$

$2$. Simplify the numerator.

=(10+2sqrt(3))/((5-sqrt(3))(5+sqrt(3))

$3$. Simplify the denominator.

$= \frac{10 + 2 \sqrt{3}}{25 - 3}$

$= \frac{10 + 2 \sqrt{3}}{22}$

$4$. Factor out $2$ from the numerator and denominator.

$= \frac{2 \left(5 + \sqrt{3}\right)}{2 \left(11\right)}$

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \left(5 + \sqrt{3}\right)}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \left(11\right)}$

$= \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \frac{5 + \sqrt{3}}{11} \textcolor{w h i t e}{\frac{a}{a}} |}}}$