Simplify 1/( square root of 2+ square root of 3)

$\frac{1}{\sqrt{2} + \sqrt{3}}$

Step 1

Multiply $\frac{1}{\sqrt{2} + \sqrt{3}}$ by $\frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}$ .

$\frac{1}{\sqrt{2} + \sqrt{3}} \cdot \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}$

Step 2

Multiply $\frac{1}{\sqrt{2} + \sqrt{3}}$ by $\frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{2} - \sqrt{3}}{{\sqrt{2}}^{2} - \sqrt{6} + \sqrt{6} - {\sqrt{3}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{2} - \sqrt{3}}{- 1}$

Step 5

Simplify the expression.

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

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

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

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

Step 6

Apply the distributive property.

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

Step 7

Multiply $- - \sqrt{3}$ .

Multiply $- 1$ by $- 1$ .

$- \sqrt{2} + 1 \sqrt{3}$

Multiply $\sqrt{3}$ by $1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.31783724 \dots$