Evaluate 5/(2+2 square root of 6)

$\frac{5}{2 + 2 \sqrt{6}}$

Step 1

Multiply $\frac{5}{2 + 2 \sqrt{6}}$ by $\frac{2 - 2 \sqrt{6}}{2 - 2 \sqrt{6}}$ .

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

Step 2

Multiply $\frac{5}{2 + 2 \sqrt{6}}$ by $\frac{2 - 2 \sqrt{6}}{2 - 2 \sqrt{6}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{5 (2 - 2 \sqrt{6})}{4 - 4 \sqrt{6} + 4 \sqrt{6} - 4 {\sqrt{6}}^{2}}$

Step 4

Simplify.

$\frac{5 (2 - 2 \sqrt{6})}{- 20}$

Step 5

Cancel the common factors.

Factor $5$ out of $- 20$ .

$\frac{5 (2 - 2 \sqrt{6})}{5 \cdot - 4}$

Cancel the common factor.

$\frac{5 (2 - 2 \sqrt{6})}{5 \cdot - 4}$

Rewrite the expression.

$\frac{2 - 2 \sqrt{6}}{- 4}$

Step 6

Cancel the common factor of $2 - 2 \sqrt{6}$ and $- 4$ .

Factor $2$ out of $2$ .

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

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

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

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

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

Cancel the common factors.

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Move the negative in front of the fraction.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.72474487 \dots$