How do you rationalize the denominator and simplify (sqrt3+7)/(8-sqrt5)?

May 1, 2018

$= \frac{56 + \sqrt{15} + 7 \sqrt{5} + 8 \sqrt{3}}{59}$

Explanation:

To rationalize the denominator, multiply by its reciprocal, which in this case is $8 + \sqrt{5}$. To use this operation you must apply it to both the top and the bottom:
$= \frac{\sqrt{3} + 7}{8 - \sqrt{5}} \cdot \frac{8 + \sqrt{5}}{8 - \sqrt{5}}$
FOIL the top and bottom:
$= \frac{8 \sqrt{3} + \sqrt{15} + 56 + 7 \sqrt{5}}{64 - 8 \sqrt{5} + 8 \sqrt{5} - 5}$
Combine all like terms:
$= \frac{56 + \sqrt{15} + 7 \sqrt{5} + 8 \sqrt{3}}{59}$
This is the simplest form.