Home » » Simplify $$\frac{1}{1 + \sqrt{5}}$$ - $$\frac{1}{1 - \sqrt{5}}$$

# Simplify $$\frac{1}{1 + \sqrt{5}}$$ - $$\frac{1}{1 - \sqrt{5}}$$

### Question

Simplify $$\frac{1}{1 + \sqrt{5}}$$ - $$\frac{1}{1 - \sqrt{5}}$$

### Options

A) - $$\frac{1}{2}\sqrt{5}$$

B) $$\frac{1}{2}\sqrt{5}$$

C) -- $$\frac{1}{4}\sqrt{5}$$

D) 5

### Explanation:

$$\frac{1}{1 + \sqrt{5}}$$ - $$\frac{1}{1 - \sqrt{5}}$$
= $$\frac{3 - \sqrt{5} - 3 - \sqrt{5}}{(3 + \sqrt{5}) (3 - \sqrt{5}}$$
= $$\frac{-2\sqrt{5}}{9 - 5}$$
= $$\frac{-2\sqrt{5}}{4}$$
= - $$\frac{1}{2}\sqrt{5}$$

## Dicussion (1)

• $$\frac{1}{1 + \sqrt{5}}$$ - $$\frac{1}{1 - \sqrt{5}}$$
= $$\frac{3 - \sqrt{5} - 3 - \sqrt{5}}{(3 + \sqrt{5}) (3 - \sqrt{5}}$$
= $$\frac{-2\sqrt{5}}{9 - 5}$$
= $$\frac{-2\sqrt{5}}{4}$$
= - $$\frac{1}{2}\sqrt{5}$$