# How do you simplify 18 /(sqrt(5) - 3 sqrt (5))?

##### 1 Answer
Apr 21, 2016

$= \frac{- 9 \sqrt{5}}{5}$

#### Explanation:

$\frac{18}{\sqrt{5} - 3 \sqrt{5}}$

Rationalizing the expression , by multiplying it with the conjugate of the denominator : color(blue)(sqrt5 + 3 sqrt5

(18 * (color(blue)(sqrt5 + 3 sqrt5) ))/ (( sqrt5 - 3 sqrt5 ) * color(blue)((sqrt5 + 3 sqrt5))

 = (18 * (color(blue)(sqrt5)) + 18 * color(blue)((3 sqrt5)) )/ (( sqrt5 - 3 sqrt5 ) * color(blue)((sqrt5 + 3 sqrt5))

• Applying property color(blue)((a-b)(a+b) = a ^2 - b^2 to the denominator.

$= \frac{18 \sqrt{5} + 54 \sqrt{5}}{\sqrt{{5}^{2}} - {\left(3 \sqrt{5}\right)}^{2}}$

$= \frac{18 \sqrt{5} + 54 \sqrt{5}}{5 - \left(9 \cdot 5\right)}$

$= \frac{18 \sqrt{5} + 54 \sqrt{5}}{5 - 45}$

$= \frac{18 \sqrt{5} + 54 \sqrt{5}}{- 40}$

$= \frac{\sqrt{5} \left(18 + 54\right)}{- 40}$

$= \frac{\sqrt{5} \left(72\right)}{- 40}$

$= \frac{\sqrt{5} \left(\cancel{72}\right)}{- \cancel{40}}$

$= \frac{\sqrt{5} \left(9\right)}{- 5}$

$= \frac{- 9 \sqrt{5}}{5}$