Simplify ( square root of 2)/(1+ square root of 6)

$\frac{\sqrt{2}}{1 + \sqrt{6}}$

Step 1

Multiply $\frac{\sqrt{2}}{1 + \sqrt{6}}$ by $\frac{1 - \sqrt{6}}{1 - \sqrt{6}}$ .

$\frac{\sqrt{2}}{1 + \sqrt{6}} \cdot \frac{1 - \sqrt{6}}{1 - \sqrt{6}}$

Step 2

Multiply $\frac{\sqrt{2}}{1 + \sqrt{6}}$ by $\frac{1 - \sqrt{6}}{1 - \sqrt{6}}$ .

$\frac{\sqrt{2} (1 - \sqrt{6})}{(1 + \sqrt{6}) (1 - \sqrt{6})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{2} (1 - \sqrt{6})}{1 - \sqrt{6} + \sqrt{6} - {\sqrt{6}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{2} (1 - \sqrt{6})}{- 5}$

Step 5

Apply the distributive property.

$\frac{\sqrt{2} \cdot 1 + \sqrt{2} (- \sqrt{6})}{- 5}$

Step 6

Multiply $\sqrt{2}$ by $1$ .

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

Step 7

Multiply $\sqrt{2} (- \sqrt{6})$ .

Combine using the product rule for radicals.

$\frac{\sqrt{2} - \sqrt{2 \cdot 6}}{- 5}$

Multiply $2$ by $6$ .

$\frac{\sqrt{2} - \sqrt{12}}{- 5}$

Step 8

Simplify each term.

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

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

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.40997761 \dots$