Simplify 3/( square root of 5+ square root of 6)

$\frac{3}{\sqrt{5} + \sqrt{6}}$

Step 1

Multiply $\frac{3}{\sqrt{5} + \sqrt{6}}$ by $\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}}$ .

$\frac{3}{\sqrt{5} + \sqrt{6}} \cdot \frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}}$

Step 2

Multiply $\frac{3}{\sqrt{5} + \sqrt{6}}$ by $\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{3 (\sqrt{5} - \sqrt{6})}{{\sqrt{5}}^{2} - \sqrt{30} + \sqrt{30} - {\sqrt{6}}^{2}}$

Step 4

Simplify.

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

Step 5

Simplify the expression.

Move the negative one from the denominator of $\frac{3 (\sqrt{5} - \sqrt{6})}{- 1}$ .

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

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

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

Step 6

Apply the distributive property.

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

Step 7

Multiply $- 1$ by $3$ .

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

Step 8

Apply the distributive property.

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

Step 9

Multiply.

Multiply $3$ by $- 1$ .

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

Multiply $- 3$ by $- 1$ .

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.64026529 \dots$