Evaluate 3/(2- square root of 5)

$\frac{3}{2 - \sqrt{5}}$

Step 1

Multiply $\frac{3}{2 - \sqrt{5}}$ by $\frac{2 + \sqrt{5}}{2 + \sqrt{5}}$ .

$\frac{3}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}}$

Step 2

Multiply $\frac{3}{2 - \sqrt{5}}$ by $\frac{2 + \sqrt{5}}{2 + \sqrt{5}}$ .

$\frac{3 (2 + \sqrt{5})}{(2 - \sqrt{5}) (2 + \sqrt{5})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{3 (2 + \sqrt{5})}{4 + 2 \sqrt{5} - 2 \sqrt{5} - {\sqrt{5}}^{2}}$

Step 4

Simplify.

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

Step 5

Simplify the expression.

Move the negative one from the denominator of $\frac{3 (2 + \sqrt{5})}{- 1}$ .

$- 1 \cdot (3 (2 + \sqrt{5}))$

Rewrite $- 1 \cdot (3 (2 + \sqrt{5}))$ as $- (3 (2 + \sqrt{5}))$ .

$- (3 (2 + \sqrt{5}))$

Step 6

Apply the distributive property.

$- (3 \cdot 2 + 3 \sqrt{5})$

Step 7

Multiply $3$ by $2$ .

$- (6 + 3 \sqrt{5})$

Step 8

Apply the distributive property.

$- 1 \cdot 6 - (3 \sqrt{5})$

Step 9

Multiply.

Multiply $- 1$ by $6$ .

$- 6 - (3 \sqrt{5})$

Multiply $3$ by $- 1$ .

$- 6 - 3 \sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 6 - 3 \sqrt{5}$

Decimal Form:

$- 12.70820393 \dots$