# How do you rationalize the denominator and simplify (3sqrt2 + 2sqrt7)/( sqrt7 - 5sqrt2)?

Mar 23, 2018

$- \frac{44 + 13 \sqrt{14}}{43}$

#### Explanation:

We first multiply the numerator and the denominator of this fraction by the conjugate of the denominator.

The denominator is $\sqrt{7} - 5 \sqrt{2}$, so its conjugate is $\sqrt{7} + 5 \sqrt{2}$, Let's do the multiplication.

$\frac{3 \sqrt{2} + 2 \sqrt{7}}{\sqrt{7} - 5 \sqrt{2}} \cdot \frac{\sqrt{7} + 5 \sqrt{2}}{\sqrt{7} + 5 \sqrt{2}}$

$\frac{\left(3 \sqrt{2} + 2 \sqrt{7}\right) \left(\sqrt{7} + 5 \sqrt{2}\right)}{\left(\sqrt{7} - 5 \sqrt{2}\right) \left(\sqrt{7} + 5 \sqrt{2}\right)}$

$\frac{3 \sqrt{14} + 15 \cdot 2 + 2 \cdot 7 + 10 \sqrt{14}}{7 + 5 \sqrt{14} - 5 \sqrt{14} - 25 \cdot 2}$

Simplifying we have

$\frac{44 + 13 \sqrt{14}}{- 43} = - \frac{44 + 13 \sqrt{14}}{43}$.