Basic Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Combine using the product rule for radicals.

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

Step 8.1.2

Multiply $3$ by $3$ .

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

Step 8.1.3

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

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

Step 8.1.4

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

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

Step 8.1.5

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

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

Step 8.1.6

Multiply $- 2$ by $- 2$ .

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

Step 8.2

Add $3$ and $4$ .

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

Step 8.3

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

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

Step 9

Apply the distributive property.

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

Step 10

Multiply $- 1$ by $7$ .

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

Step 11

Multiply $- 4$ by $- 1$ .

$- 7 + 4 \sqrt{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- 7 + 4 \sqrt{3}$

Decimal Form:

$- 0.07179676 \dots$