# How do you simplify (1+2sqrt3)/(1+sqrt3)?

Jul 7, 2015

=color(blue)(( 5-sqrt3)/(2)

#### Explanation:

$\frac{1 + 2 \sqrt{3}}{1 + \sqrt{3}}$

Multiplying the numerator and denominator by the conjugate
$\textcolor{b l u e}{1 - \sqrt{3}}$

$= \frac{\left(1 + 2 \sqrt{3}\right) . \left(\textcolor{b l u e}{1 - \sqrt{3}}\right)}{\left(1 + \sqrt{3}\right) . \textcolor{b l u e}{\left(1 - \sqrt{3}\right)}}$

$= \frac{\left(1 - \sqrt{3} + 2 \sqrt{3} - 2 \sqrt{3} . \sqrt{3}\right)}{{1}^{2} - {\left(\sqrt{3}\right)}^{2}}$

$= \frac{\left(1 + \sqrt{3} - 6\right)}{1 - 3}$

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

=color(blue)((( 5-sqrt3))/(2)