Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{5 + 2 \sqrt{6}} (5 + 2 \sqrt{6})}{25 + 10 \sqrt{6} - 10 \sqrt{6} - 4 {\sqrt{6}}^{2}}$

Step 4

Simplify.

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

Step 5

Divide $\sqrt{5 + 2 \sqrt{6}} (5 + 2 \sqrt{6})$ by $1$ .

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

Step 6

Apply the distributive property.

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

Step 7

Move $5$ to the left of $\sqrt{5 + 2 \sqrt{6}}$ .

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

Step 8

Combine using the product rule for radicals.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$31.14480645 \dots$