How do you simplify 1/(3+sqrt5) ?

Apr 7, 2018

$\frac{3 - \sqrt{5}}{4}$

Explanation:

Rationalize the denominator (radicals cannot be in the denominator) by multiplying the numerator and denominator by $3 - \sqrt{5}$

$\frac{1}{3 + \sqrt{5}}$

$\frac{1 \cdot 3 - \sqrt{5}}{\left(3 + \sqrt{5}\right) \left(3 - \sqrt{5}\right)}$

$\left(3 + \sqrt{5}\right) \left(3 - \sqrt{5}\right) = 9 - 3 \sqrt{5} + 3 \sqrt{5} - \sqrt{25}$

$9 - 3 \sqrt{5} + 3 \sqrt{5} - \sqrt{25} = 9 - 5 = 4$

$\frac{3 - \sqrt{5}}{4}$

Apr 7, 2018

$\frac{3 - \sqrt{5}}{4}$

Explanation:

You can rationalise the denominator by multiplying by it’s conjugate.

$\frac{1}{3 + \sqrt{5}} \times \frac{3 - \sqrt{5}}{3 - \sqrt{5}}$

$= \frac{3 - \sqrt{5}}{9 - 5}$

$= \frac{3 - \sqrt{5}}{4}$