Home » » Simplify $$\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$$

# Simplify $$\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$$

### Question

Simplify $$\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$$

### Options

A) 3$$\sqrt{6} - 7$$

B) 3$$\sqrt{6} - 7$$

C) 3$$\sqrt{6} - 1$$

D) 3$$\sqrt{6} + 1$$

### Explanation:

$$= \frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$$
$$= \frac{2\sqrt{2}(\sqrt{2}) + (2\sqrt{2})(-\sqrt{3})-\sqrt{3}(\sqrt{2})-\sqrt{3}(-\sqrt{3})}{(\sqrt{2})^2-(\sqrt{3})^2}$$
$$= \frac{2 \times 2 - 2\sqrt{6} - \sqrt{6} + 3}{2 - 3}$$
$$= \frac{4 - 3\sqrt{6} + 3}{-1}$$
$$= \frac{7 - 3\sqrt{6}}{-1}$$
$$= \frac{7}{-1} - \frac{3\sqrt{6}}{-1}$$
$$= -7 + 3\sqrt{6}$$
$$= 3\sqrt{6}-7$$

## Dicussion (1)

• $$= \frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$$
$$= \frac{2\sqrt{2}(\sqrt{2}) + (2\sqrt{2})(-\sqrt{3})-\sqrt{3}(\sqrt{2})-\sqrt{3}(-\sqrt{3})}{(\sqrt{2})^2-(\sqrt{3})^2}$$
$$= \frac{2 \times 2 - 2\sqrt{6} - \sqrt{6} + 3}{2 - 3}$$
$$= \frac{4 - 3\sqrt{6} + 3}{-1}$$
$$= \frac{7 - 3\sqrt{6}}{-1}$$
$$= \frac{7}{-1} - \frac{3\sqrt{6}}{-1}$$
$$= -7 + 3\sqrt{6}$$
$$= 3\sqrt{6}-7$$