Simplify ( square root of 2)÷(2+ square root of 3)

$\sqrt{2} \div (2 + \sqrt{3})$

Step 1

Rewrite the division as a fraction.

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

Step 2

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

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

Step 3

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

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

Step 4

Expand the denominator using the FOIL method.

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

Step 5

Simplify.

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

Step 6

Divide $\sqrt{2} (2 - \sqrt{3})$ by $1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Move $2$ to the left of $\sqrt{2}$ .

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

Step 9

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$2 \cdot \sqrt{2} - \sqrt{6}$

$2 \sqrt{2} - \sqrt{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2} - \sqrt{6}$

Decimal Form:

$0.37893738 \dots$