Algebra Examples

$\frac{8}{2 - \sqrt{12}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1.1

Factor $4$ out of $12$ .

$\frac{8}{2 - \sqrt{4 (3)}}$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{8}{2 - \sqrt{2^{2} \cdot 3}}$

Step 1.2

Pull terms out from under the radical.

$\frac{8}{2 - (2 \sqrt{3})}$

Step 1.3

Multiply $2$ by $- 1$ .

$\frac{8}{2 - 2 \sqrt{3}}$

Step 2

Cancel the common factor of $8$ and $2 - 2 \sqrt{3}$ .

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Step 2.1

Factor $2$ out of $8$ .

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $2$ .

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

Step 2.2.2

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Step 2.2.3

Factor $2$ out of $2 (1) + 2 (- \sqrt{3})$ .

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

Step 2.2.4

Cancel the common factor.

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

Step 2.2.5

Rewrite the expression.

$\frac{4}{1 - \sqrt{3}}$

Step 3

Multiply $\frac{4}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

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

Step 4

Multiply $\frac{4}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

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

Step 5

Expand the denominator using the FOIL method.

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

Step 6

Simplify.

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

Step 7

Cancel the common factor of $4$ and $- 2$ .

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Step 7.1

Factor $2$ out of $4 (1 + \sqrt{3})$ .

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

Step 7.2

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

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

Step 8

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

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

Step 9

Apply the distributive property.

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

Step 10

Multiply $2$ by $1$ .

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

Step 11

Apply the distributive property.

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

Step 12

Multiply.

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Step 12.1

Multiply $- 1$ by $2$ .

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

Step 12.2

Multiply $2$ by $- 1$ .

$- 2 - 2 \sqrt{3}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- 2 - 2 \sqrt{3}$

Decimal Form:

$- 5.46410161 \dots$