# How do you simplify  (5-3sqrt6)/(3+4sqrt2)?

Apr 2, 2017

$\frac{9 \sqrt{6} + 20 \sqrt{2} - 24 \sqrt{3} - 15}{23}$

#### Explanation:

In order to rationalize the denominator, we multiply it by its conjugate (the conjugate of $a + b \sqrt{c}$ is $a - b \sqrt{c}$). The reason why will be apparent as we solve the problem.

The conjugate of the denominator $3 + 4 \sqrt{2}$ is $3 - 4 \sqrt{2}$.

Since we are going to multiply its denominator by $3 - 4 \sqrt{2}$, we also have to multiply the numerator by $3 - 4 \sqrt{2}$.

Then, we have $\frac{5 - 3 \sqrt{6}}{3 + 4 \sqrt{2}} = \frac{5 - 3 \sqrt{6}}{3 + 4 \sqrt{2}} \cdot \frac{3 - 4 \sqrt{2}}{3 - 4 \sqrt{2}} = \frac{\left(5 - 3 \sqrt{6}\right) \left(3 - 4 \sqrt{2}\right)}{\left(3 + 4 \sqrt{2}\right) \left(3 - 4 \sqrt{2}\right)}$.

We can simplify the denominator by using the identity $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$.

$\frac{\left(5 - 3 \sqrt{6}\right) \left(3 - 4 \sqrt{2}\right)}{\left(3 + 4 \sqrt{2}\right) \left(3 - 4 \sqrt{2}\right)} = \frac{15 - 9 \sqrt{6} - 20 \sqrt{2} + 12 \sqrt{12}}{9 - 32} = - \frac{15 - 9 \sqrt{6} - 20 \sqrt{2} + 24 \sqrt{3}}{23} = \frac{9 \sqrt{6} + 20 \sqrt{2} - 24 \sqrt{3} - 15}{23}$

Apr 2, 2017

$- \frac{15 - 9 \sqrt{6} - 20 \sqrt{2} + 24 \sqrt{3}}{23}$

#### Explanation:

We can simplify the denominator by turning it into a single term by multiplying the fraction by the conjugate of the denominator.

The conjugate of $3 + 4 \sqrt{2}$ is found by simply reversing the sign of the second term, which is $3 - 4 \sqrt{2}$.

So, we multiply the numerator and denominator by $3 - 4 \sqrt{2}$.

$\frac{5 - 3 \sqrt{6}}{3 + 4 \sqrt{2}} \cdot \frac{3 - 4 \sqrt{2}}{3 - 4 \sqrt{2}} = \frac{\left(5 - 3 \sqrt{6}\right) \left(3 - 4 \sqrt{2}\right)}{\left(3 + 4 \sqrt{2}\right) \left(3 - 4 \sqrt{2}\right)}$

Expand both of these by FOILing:

$= \frac{15 - \left(3 \sqrt{6}\right) 3 + 5 \left(- 4 \sqrt{2}\right) - 3 \sqrt{6} \left(- 4 \sqrt{2}\right)}{9 + \left(4 \sqrt{2}\right) 3 + 3 \left(- 4 \sqrt{2}\right) + \left(4 \sqrt{2}\right) \left(- 4 \sqrt{2}\right)}$

Simplifying these:

$= \frac{15 - 9 \sqrt{6} - 20 \sqrt{2} + 12 \sqrt{12}}{9 + 12 \sqrt{2} - 12 \sqrt{2} - 16 \sqrt{4}}$

Note that $\sqrt{4} = 2$ and $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \sqrt{3} = 2 \sqrt{3}$.

$= \frac{15 - 9 \sqrt{6} - 20 \sqrt{2} + 12 \left(2 \sqrt{3}\right)}{9 - 16 \left(2\right)}$

$= \frac{15 - 9 \sqrt{6} - 20 \sqrt{2} + 24 \sqrt{3}}{9 - 32}$

$= \frac{15 - 9 \sqrt{6} - 20 \sqrt{2} + 24 \sqrt{3}}{- 23}$