# How do you rationalize the denominator and simplify (1+sqrt5)/(10+4sqrt15)?

Jun 12, 2018

$\frac{- 5 + 2 \sqrt{15} - 5 \sqrt{5} + 10 \sqrt{3}}{70}$

#### Explanation:

$\frac{1 + \sqrt{5}}{10 + 4 \sqrt{15}} \cdot \frac{10 - 4 \sqrt{15}}{10 - 4 \sqrt{15}}$

$= \frac{10 - 4 \sqrt{15} + 10 \sqrt{5} - 4 \cdot 5 \sqrt{3}}{100 - 16 \cdot 15}$

Jun 12, 2018

$\frac{5 - 2 \sqrt{15} + 5 \sqrt{5} - 2 \sqrt{75}}{- 70}$

#### Explanation:

$\frac{1 + \sqrt{5}}{10 + 4 \sqrt{15}}$

Multiply by the conjugate which is the denominator with the sign changed over itself:

$\frac{10 - 4 \sqrt{15}}{10 - 4 \sqrt{15}}$

$\frac{1 + \sqrt{5}}{10 + 4 \sqrt{15}} \cdot \frac{10 - 4 \sqrt{15}}{10 - 4 \sqrt{15}}$

FOIL:

$\frac{10 - 4 \sqrt{15} + 10 \sqrt{5} - 4 \sqrt{75}}{100 - 40 \sqrt{15} + 40 \sqrt{15} - 16 \cdot 15}$

$\frac{5 - 2 \sqrt{15} + 5 \sqrt{5} - 2 \sqrt{75}}{- 70}$