Simplify ( square root of 3-2)/( square root of 3+2)+( square root of 3+2)/( square root of 3-2)

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

Step 1

Simplify each term.

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Move the negative one from the denominator of $\frac{(\sqrt{3} - 2) (\sqrt{3} - 2)}{- 1}$ .

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

Rewrite $- 1 \cdot ((\sqrt{3} - 2) (\sqrt{3} - 2))$ as $- ((\sqrt{3} - 2) (\sqrt{3} - 2))$ .

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

Expand $(\sqrt{3} - 2) (\sqrt{3} - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

$- (\sqrt{9} + \sqrt{3} \cdot - 2 - 2 \sqrt{3} - 2 \cdot - 2) + \frac{\sqrt{3} + 2}{\sqrt{3} - 2}$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Move $- 2$ to the left of $\sqrt{3}$ .

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

Multiply $- 2$ by $- 2$ .

$- (3 - 2 \sqrt{3} - 2 \sqrt{3} + 4) + \frac{\sqrt{3} + 2}{\sqrt{3} - 2}$

Add $3$ and $4$ .

$- (7 - 2 \sqrt{3} - 2 \sqrt{3}) + \frac{\sqrt{3} + 2}{\sqrt{3} - 2}$

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$- (7 - 4 \sqrt{3}) + \frac{\sqrt{3} + 2}{\sqrt{3} - 2}$

Apply the distributive property.

$- 1 \cdot 7 - (- 4 \sqrt{3}) + \frac{\sqrt{3} + 2}{\sqrt{3} - 2}$

Multiply $- 1$ by $7$ .

$- 7 - (- 4 \sqrt{3}) + \frac{\sqrt{3} + 2}{\sqrt{3} - 2}$

Multiply $- 4$ by $- 1$ .

$- 7 + 4 \sqrt{3} + \frac{\sqrt{3} + 2}{\sqrt{3} - 2}$

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

$- 7 + 4 \sqrt{3} + \frac{\sqrt{3} + 2}{\sqrt{3} - 2} \cdot \frac{\sqrt{3} + 2}{\sqrt{3} + 2}$

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

$- 7 + 4 \sqrt{3} + \frac{(\sqrt{3} + 2) (\sqrt{3} + 2)}{(\sqrt{3} - 2) (\sqrt{3} + 2)}$

Expand the denominator using the FOIL method.

$- 7 + 4 \sqrt{3} + \frac{(\sqrt{3} + 2) (\sqrt{3} + 2)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot 2 - 2 \sqrt{3} - 4}$

Simplify.

$- 7 + 4 \sqrt{3} + \frac{(\sqrt{3} + 2) (\sqrt{3} + 2)}{- 1}$

Move the negative one from the denominator of $\frac{(\sqrt{3} + 2) (\sqrt{3} + 2)}{- 1}$ .

$- 7 + 4 \sqrt{3} - 1 \cdot ((\sqrt{3} + 2) (\sqrt{3} + 2))$

Rewrite $- 1 \cdot ((\sqrt{3} + 2) (\sqrt{3} + 2))$ as $- ((\sqrt{3} + 2) (\sqrt{3} + 2))$ .

$- 7 + 4 \sqrt{3} - ((\sqrt{3} + 2) (\sqrt{3} + 2))$

Expand $(\sqrt{3} + 2) (\sqrt{3} + 2)$ using the FOIL Method.

Apply the distributive property.

$- 7 + 4 \sqrt{3} - (\sqrt{3} (\sqrt{3} + 2) + 2 (\sqrt{3} + 2))$

Apply the distributive property.

$- 7 + 4 \sqrt{3} - (\sqrt{3} \sqrt{3} + \sqrt{3} \cdot 2 + 2 (\sqrt{3} + 2))$

Apply the distributive property.

$- 7 + 4 \sqrt{3} - (\sqrt{3} \sqrt{3} + \sqrt{3} \cdot 2 + 2 \sqrt{3} + 2 \cdot 2)$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$- 7 + 4 \sqrt{3} - (\sqrt{3 \cdot 3} + \sqrt{3} \cdot 2 + 2 \sqrt{3} + 2 \cdot 2)$

Multiply $3$ by $3$ .

$- 7 + 4 \sqrt{3} - (\sqrt{9} + \sqrt{3} \cdot 2 + 2 \sqrt{3} + 2 \cdot 2)$

Rewrite $9$ as $3^{2}$ .

$- 7 + 4 \sqrt{3} - (\sqrt{3^{2}} + \sqrt{3} \cdot 2 + 2 \sqrt{3} + 2 \cdot 2)$

Pull terms out from under the radical, assuming positive real numbers.

$- 7 + 4 \sqrt{3} - (3 + \sqrt{3} \cdot 2 + 2 \sqrt{3} + 2 \cdot 2)$

Move $2$ to the left of $\sqrt{3}$ .

$- 7 + 4 \sqrt{3} - (3 + 2 \cdot \sqrt{3} + 2 \sqrt{3} + 2 \cdot 2)$

Multiply $2$ by $2$ .

$- 7 + 4 \sqrt{3} - (3 + 2 \sqrt{3} + 2 \sqrt{3} + 4)$

Add $3$ and $4$ .

$- 7 + 4 \sqrt{3} - (7 + 2 \sqrt{3} + 2 \sqrt{3})$

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$- 7 + 4 \sqrt{3} - (7 + 4 \sqrt{3})$

Apply the distributive property.

$- 7 + 4 \sqrt{3} - 1 \cdot 7 - (4 \sqrt{3})$

Multiply $- 1$ by $7$ .

$- 7 + 4 \sqrt{3} - 7 - (4 \sqrt{3})$

Multiply $4$ by $- 1$ .

$- 7 + 4 \sqrt{3} - 7 - 4 \sqrt{3}$

Step 2

Simplify by adding terms.

Subtract $7$ from $- 7$ .

$- 14 + 4 \sqrt{3} - 4 \sqrt{3}$

Subtract $4 \sqrt{3}$ from $4 \sqrt{3}$ .

$- 14 + 0$

Add $- 14$ and $0$ .

$- 14$

Simplify ( square root of 3-2)/( square root of 3+2)+( square root of 3+2)/( square root of 3-2)

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