Evaluate 1/( square root of 5-2)-1/( square root of 5+2)

$\frac{1}{\sqrt{5} - 2} - \frac{1}{\sqrt{5} + 2}$

Step 1

Simplify each term.

Multiply $\frac{1}{\sqrt{5} - 2}$ by $\frac{\sqrt{5} + 2}{\sqrt{5} + 2}$ .

$\frac{1}{\sqrt{5} - 2} \cdot \frac{\sqrt{5} + 2}{\sqrt{5} + 2} - \frac{1}{\sqrt{5} + 2}$

Multiply $\frac{1}{\sqrt{5} - 2}$ by $\frac{\sqrt{5} + 2}{\sqrt{5} + 2}$ .

$\frac{\sqrt{5} + 2}{(\sqrt{5} - 2) (\sqrt{5} + 2)} - \frac{1}{\sqrt{5} + 2}$

Expand the denominator using the FOIL method.

$\frac{\sqrt{5} + 2}{{\sqrt{5}}^{2} + \sqrt{5} \cdot 2 - 2 \sqrt{5} - 4} - \frac{1}{\sqrt{5} + 2}$

Simplify.

$\frac{\sqrt{5} + 2}{1} - \frac{1}{\sqrt{5} + 2}$

Divide $\sqrt{5} + 2$ by $1$ .

$\sqrt{5} + 2 - \frac{1}{\sqrt{5} + 2}$

Multiply $\frac{1}{\sqrt{5} + 2}$ by $\frac{\sqrt{5} - 2}{\sqrt{5} - 2}$ .

$\sqrt{5} + 2 - (\frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2})$

Multiply $\frac{1}{\sqrt{5} + 2}$ by $\frac{\sqrt{5} - 2}{\sqrt{5} - 2}$ .

$\sqrt{5} + 2 - \frac{\sqrt{5} - 2}{(\sqrt{5} + 2) (\sqrt{5} - 2)}$

Expand the denominator using the FOIL method.

$\sqrt{5} + 2 - \frac{\sqrt{5} - 2}{{\sqrt{5}}^{2} + \sqrt{5} \cdot - 2 + 2 \sqrt{5} - 4}$

Simplify.

$\sqrt{5} + 2 - \frac{\sqrt{5} - 2}{1}$

Divide $\sqrt{5} - 2$ by $1$ .

$\sqrt{5} + 2 - (\sqrt{5} - 2)$

Apply the distributive property.

$\sqrt{5} + 2 - \sqrt{5} - - 2$

Multiply $- 1$ by $- 2$ .

$\sqrt{5} + 2 - \sqrt{5} + 2$

Step 2

Simplify by adding terms.

Subtract $\sqrt{5}$ from $\sqrt{5}$ .

$0 + 2 + 2$

Simplify by adding numbers.

Add $0$ and $2$ .

$2 + 2$

Add $2$ and $2$ .

$4$