Algebra Examples

$\frac{2 \sqrt{15}}{- 4 - \sqrt{12}}$

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 $12$ .

$\frac{2 \sqrt{15}}{- 4 - \sqrt{4 (3)}}$

Step 1.1.2

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

$\frac{2 \sqrt{15}}{- 4 - \sqrt{2^{2} \cdot 3}}$

Step 1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{15}}{- 4 - (2 \sqrt{3})}$

Step 1.3

Multiply $2$ by $- 1$ .

$\frac{2 \sqrt{15}}{- 4 - 2 \sqrt{3}}$

Step 2

Cancel the common factor of $2$ and $- 4 - 2 \sqrt{3}$ .

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Step 2.1

Factor $2$ out of $2 \sqrt{15}$ .

$\frac{2 (\sqrt{15})}{- 4 - 2 \sqrt{3}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $- 4$ .

$\frac{2 (\sqrt{15})}{2 (- 2) - 2 \sqrt{3}}$

Step 2.2.2

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

$\frac{2 (\sqrt{15})}{2 (- 2) + 2 (- \sqrt{3})}$

Step 2.2.3

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

$\frac{2 (\sqrt{15})}{2 (- 2 - \sqrt{3})}$

Step 2.2.4

Cancel the common factor.

$\frac{2 \sqrt{15}}{2 (- 2 - \sqrt{3})}$

Step 2.2.5

Rewrite the expression.

$\frac{\sqrt{15}}{- 2 - \sqrt{3}}$

Step 3

Multiply $\frac{\sqrt{15}}{- 2 - \sqrt{3}}$ by $\frac{- 2 + \sqrt{3}}{- 2 + \sqrt{3}}$ .

$\frac{\sqrt{15}}{- 2 - \sqrt{3}} \cdot \frac{- 2 + \sqrt{3}}{- 2 + \sqrt{3}}$

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{\sqrt{15}}{- 2 - \sqrt{3}}$ by $\frac{- 2 + \sqrt{3}}{- 2 + \sqrt{3}}$ .

$\frac{\sqrt{15} (- 2 + \sqrt{3})}{(- 2 - \sqrt{3}) (- 2 + \sqrt{3})}$

Step 4.2

Expand the denominator using the FOIL method.

$\frac{\sqrt{15} (- 2 + \sqrt{3})}{4 - 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 4.3

Simplify.

$\frac{\sqrt{15} (- 2 + \sqrt{3})}{1}$

Step 4.4

Divide $\sqrt{15} (- 2 + \sqrt{3})$ by $1$ .

$\sqrt{15} (- 2 + \sqrt{3})$

Step 4.5

Apply the distributive property.

$\sqrt{15} \cdot - 2 + \sqrt{15} \sqrt{3}$

Step 4.6

Move $- 2$ to the left of $\sqrt{15}$ .

$- 2 \cdot \sqrt{15} + \sqrt{15} \sqrt{3}$

Step 4.7

Combine using the product rule for radicals.

$- 2 \cdot \sqrt{15} + \sqrt{15 \cdot 3}$

Step 5

Simplify each term.

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Step 5.1

Multiply $15$ by $3$ .

$- 2 \sqrt{15} + \sqrt{45}$

Step 5.2

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 5.2.1

Factor $9$ out of $45$ .

$- 2 \sqrt{15} + \sqrt{9 (5)}$

Step 5.2.2

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

$- 2 \sqrt{15} + \sqrt{3^{2} \cdot 5}$

Step 5.3

Pull terms out from under the radical.

$- 2 \sqrt{15} + 3 \sqrt{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- 2 \sqrt{15} + 3 \sqrt{5}$

Decimal Form:

$- 1.03776275 \dots$