Find the Exact Value 1/(2+ square root of 3)+1/(2- square root of 3)

$\frac{1}{2 + \sqrt{3}} + \frac{1}{2 - \sqrt{3}}$

Step 1

Simplify each term.

Multiply $\frac{1}{2 + \sqrt{3}}$ by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ .

$\frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} + \frac{1}{2 - \sqrt{3}}$

Multiply $\frac{1}{2 + \sqrt{3}}$ by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ .

$\frac{2 - \sqrt{3}}{(2 + \sqrt{3}) (2 - \sqrt{3})} + \frac{1}{2 - \sqrt{3}}$

Expand the denominator using the FOIL method.

$\frac{2 - \sqrt{3}}{4 - 2 \sqrt{3} + \sqrt{3} \cdot 2 - {\sqrt{3}}^{2}} + \frac{1}{2 - \sqrt{3}}$

Simplify.

$\frac{2 - \sqrt{3}}{1} + \frac{1}{2 - \sqrt{3}}$

Divide $2 - \sqrt{3}$ by $1$ .

$2 - \sqrt{3} + \frac{1}{2 - \sqrt{3}}$

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

$2 - \sqrt{3} + \frac{1}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}}$

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

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

Expand the denominator using the FOIL method.

$2 - \sqrt{3} + \frac{2 + \sqrt{3}}{4 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{3}}^{2}}$

Simplify.

$2 - \sqrt{3} + \frac{2 + \sqrt{3}}{1}$

Divide $2 + \sqrt{3}$ by $1$ .

$2 - \sqrt{3} + 2 + \sqrt{3}$

Step 2

Simplify by adding terms.

Add $2$ and $2$ .

$4 - \sqrt{3} + \sqrt{3}$

Add $- \sqrt{3}$ and $\sqrt{3}$ .

$4 + 0$

Add $4$ and $0$ .

$4$