# How do you simplify 2div( 5 - sqrt3)?

May 1, 2018

Multiply denominator and numerator with $5 + \sqrt{3}$

#### Explanation:

Remember that (a+b)(a-b) =${a}^{2} - {b}^{2}$
That gives you
$\frac{2}{5 - \sqrt{3}}$
=$\frac{2 \left(5 + \sqrt{3}\right)}{\left(5 + \sqrt{3}\right) \left(5 - \sqrt{3}\right)}$
= $\frac{2 \left(5 + \sqrt{3}\right)}{25 - 9}$
= $\frac{5 + \sqrt{3}}{8}$

May 1, 2018

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

#### Explanation:

= 2/(5-sqrt(3)
We multiply and divide the fraction by the denominator's conjugate to eliminate the irrationality in the denominator.

$= \frac{2}{5 - \sqrt{3}} \times \frac{5 + \sqrt{3}}{5 + \sqrt{3}}$
Using $\left(a - b\right) \left(a + b\right) = {a}^{2} - {b}^{2}$, we have
$= \frac{2 \left(5 + \sqrt{3}\right)}{22}$
$= \frac{5 + \sqrt{3}}{11}$

May 1, 2018

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

#### Explanation:

To rationalize this expression, multiply both sides by the bottom's inverse $\left(5 + \sqrt{3}\right)$
$\frac{2}{5 - \sqrt{3}} \cdot \frac{5 + \sqrt{3}}{5 + \sqrt{3}}$ Distribute:
$= \frac{10 + 2 \sqrt{3}}{25 + 5 \sqrt{3} - 5 \sqrt{3} - 3}$ Combine like terms:
$= \frac{10 + 2 \sqrt{3}}{22}$ Divide by $2$:
$= \frac{5 + \sqrt{3}}{11}$ Simplest form.