Simplify 1/(11+2 square root of 30)+1/(11-2 square root of 30)

$\frac{1}{11 + 2 \sqrt{30}} + \frac{1}{11 - 2 \sqrt{30}}$

Step 1

Simplify each term.

Multiply $\frac{1}{11 + 2 \sqrt{30}}$ by $\frac{11 - 2 \sqrt{30}}{11 - 2 \sqrt{30}}$ .

$\frac{1}{11 + 2 \sqrt{30}} \cdot \frac{11 - 2 \sqrt{30}}{11 - 2 \sqrt{30}} + \frac{1}{11 - 2 \sqrt{30}}$

Multiply $\frac{1}{11 + 2 \sqrt{30}}$ by $\frac{11 - 2 \sqrt{30}}{11 - 2 \sqrt{30}}$ .

$\frac{11 - 2 \sqrt{30}}{(11 + 2 \sqrt{30}) (11 - 2 \sqrt{30})} + \frac{1}{11 - 2 \sqrt{30}}$

Expand the denominator using the FOIL method.

$\frac{11 - 2 \sqrt{30}}{121 - 22 \sqrt{30} + 22 \sqrt{30} - 4 {\sqrt{30}}^{2}} + \frac{1}{11 - 2 \sqrt{30}}$

Simplify.

$\frac{11 - 2 \sqrt{30}}{1} + \frac{1}{11 - 2 \sqrt{30}}$

Divide $11 - 2 \sqrt{30}$ by $1$ .

$11 - 2 \sqrt{30} + \frac{1}{11 - 2 \sqrt{30}}$

Multiply $\frac{1}{11 - 2 \sqrt{30}}$ by $\frac{11 + 2 \sqrt{30}}{11 + 2 \sqrt{30}}$ .

$11 - 2 \sqrt{30} + \frac{1}{11 - 2 \sqrt{30}} \cdot \frac{11 + 2 \sqrt{30}}{11 + 2 \sqrt{30}}$

Multiply $\frac{1}{11 - 2 \sqrt{30}}$ by $\frac{11 + 2 \sqrt{30}}{11 + 2 \sqrt{30}}$ .

$11 - 2 \sqrt{30} + \frac{11 + 2 \sqrt{30}}{(11 - 2 \sqrt{30}) (11 + 2 \sqrt{30})}$

Expand the denominator using the FOIL method.

$11 - 2 \sqrt{30} + \frac{11 + 2 \sqrt{30}}{121 + 22 \sqrt{30} - 22 \sqrt{30} - 4 {\sqrt{30}}^{2}}$

Simplify.

$11 - 2 \sqrt{30} + \frac{11 + 2 \sqrt{30}}{1}$

Divide $11 + 2 \sqrt{30}$ by $1$ .

$11 - 2 \sqrt{30} + 11 + 2 \sqrt{30}$

Step 2

Simplify by adding terms.

Add $11$ and $11$ .

$22 - 2 \sqrt{30} + 2 \sqrt{30}$

Add $- 2 \sqrt{30}$ and $2 \sqrt{30}$ .

$22 + 0$

Add $22$ and $0$ .

$22$