Basic Math Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Expand the denominator using the FOIL method.

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

Step 1.4

Simplify.

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

Tap for more steps...

Step 1.7.1

Apply the distributive property.

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Apply the distributive property.

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

Step 1.8

Simplify and combine like terms.

Tap for more steps...

Step 1.8.1

Simplify each term.

Tap for more steps...

Step 1.8.1.1

Combine using the product rule for radicals.

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

Step 1.8.1.2

Multiply $3$ by $3$ .

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

Step 1.8.1.3

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

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

Step 1.8.1.4

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

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

Step 1.8.1.5

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

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

Step 1.8.1.6

Multiply $- 2$ by $- 2$ .

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

Step 1.8.2

Add $3$ and $4$ .

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

Step 1.8.3

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

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

Step 1.9

Apply the distributive property.

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

Step 1.10

Multiply $- 1$ by $7$ .

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

Step 1.11

Multiply $- 4$ by $- 1$ .

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

Step 1.12

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

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

Step 1.13

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

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

Step 1.14

Expand the denominator using the FOIL method.

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

Step 1.15

Simplify.

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

Step 1.16

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

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

Step 1.17

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

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

Step 1.18

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

Tap for more steps...

Step 1.18.1

Apply the distributive property.

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

Step 1.18.2

Apply the distributive property.

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

Step 1.18.3

Apply the distributive property.

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

Step 1.19

Simplify and combine like terms.

Tap for more steps...

Step 1.19.1

Simplify each term.

Tap for more steps...

Step 1.19.1.1

Combine using the product rule for radicals.

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

Step 1.19.1.2

Multiply $3$ by $3$ .

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

Step 1.19.1.3

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

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

Step 1.19.1.4

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

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

Step 1.19.1.5

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

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

Step 1.19.1.6

Multiply $2$ by $2$ .

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

Step 1.19.2

Add $3$ and $4$ .

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

Step 1.19.3

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

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

Step 1.20

Apply the distributive property.

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

Step 1.21

Multiply $- 1$ by $7$ .

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

Step 1.22

Multiply $4$ by $- 1$ .

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

Step 2

Simplify by adding terms.

Tap for more steps...

Step 2.1

Subtract $7$ from $- 7$ .

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

Step 2.2

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

$- 14 + 0$

Step 2.3

Add $- 14$ and $0$ .

$- 14$