# How do you simplify (sqrt(3) - 2) / (-5*sqrt(3)+8)?

Jun 12, 2017

color(green)((1+2sqrt3)/11

#### Explanation:

$\frac{\sqrt{3} - 2}{- 5 \cdot \sqrt{3} + 8}$

$\frac{\sqrt{3} - 2}{- 5 \cdot \sqrt{3} + 8} \times \frac{- 5 \sqrt{3} - 8}{- 5 \sqrt{3} - 8}$ (conjugate)

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\sqrt{3} - 2$
$\textcolor{w h i t e}{a a a a a a a a}$$\times \underline{- 5 \sqrt{3} - 8}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\left(- 5 \cdot 3\right) + 10 \sqrt{3}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a}$$- 8 \sqrt{3} + 16$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a}$$\overline{- 15 + 2 \sqrt{3} + 16}$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$color(green)(1+2sqrt3 numerator

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$- 5 \sqrt{3} + 8$
$\textcolor{w h i t e}{a a a a a a a a a a a}$$\times \underline{- 5 \sqrt{3} - 8}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\left(25 \cdot 3\right) - 40 \sqrt{3}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a}$$\underline{+ 40 \sqrt{3} - 64}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a}$$75 - 64$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$color(green)(11 denominator

:.color(green)(=(1+2sqrt3)/11

Check:

color(red)((sqrt3-2)/(-5*sqrt3+8)=0.405827418

color(red)((1+2sqrt3)/11=0.405827418