Evaluate 8/(2+2 square root of 6)

$\frac{8}{2 + 2 \sqrt{6}}$

Step 1

Cancel the common factor of $8$ and $2 + 2 \sqrt{6}$ .

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2 + 2 \sqrt{6}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 \cdot 4}{2 (1) + 2 \sqrt{6}}$

Factor $2$ out of $2 \sqrt{6}$ .

$\frac{2 \cdot 4}{2 (1) + 2 (\sqrt{6})}$

Factor $2$ out of $2 (1) + 2 (\sqrt{6})$ .

$\frac{2 \cdot 4}{2 (1 + \sqrt{6})}$

Cancel the common factor.

$\frac{2 \cdot 4}{2 (1 + \sqrt{6})}$

Rewrite the expression.

$\frac{4}{1 + \sqrt{6}}$

Step 2

Multiply $\frac{4}{1 + \sqrt{6}}$ by $\frac{1 - \sqrt{6}}{1 - \sqrt{6}}$ .

$\frac{4}{1 + \sqrt{6}} \cdot \frac{1 - \sqrt{6}}{1 - \sqrt{6}}$

Step 3

Multiply $\frac{4}{1 + \sqrt{6}}$ by $\frac{1 - \sqrt{6}}{1 - \sqrt{6}}$ .

$\frac{4 (1 - \sqrt{6})}{(1 + \sqrt{6}) (1 - \sqrt{6})}$

Step 4

Expand the denominator using the FOIL method.

$\frac{4 (1 - \sqrt{6})}{1 - \sqrt{6} + \sqrt{6} - {\sqrt{6}}^{2}}$

Step 5

Simplify.

$\frac{4 (1 - \sqrt{6})}{- 5}$

Step 6

Move the negative in front of the fraction.

$- \frac{4 (1 - \sqrt{6})}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 (1 - \sqrt{6})}{5}$

Decimal Form:

$1.15959179 \dots$