Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(- 2 + \sqrt{3}) (\sqrt{3} + \sqrt{15})}{{\sqrt{3}}^{2} + \sqrt{45} - \sqrt{45} - {\sqrt{15}}^{2}}$

Step 2.3

Simplify.

$\frac{(- 2 + \sqrt{3}) (\sqrt{3} + \sqrt{15})}{- 12}$

Step 3

Expand $(- 2 + \sqrt{3}) (\sqrt{3} + \sqrt{15})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{- 2 (\sqrt{3} + \sqrt{15}) + \sqrt{3} (\sqrt{3} + \sqrt{15})}{- 12}$

Step 3.2

Apply the distributive property.

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + \sqrt{3} (\sqrt{3} + \sqrt{15})}{- 12}$

Step 3.3

Apply the distributive property.

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + \sqrt{3} \sqrt{3} + \sqrt{3} \sqrt{15}}{- 12}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine using the product rule for radicals.

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + \sqrt{3 \cdot 3} + \sqrt{3} \sqrt{15}}{- 12}$

Step 4.1.2

Multiply $3$ by $3$ .

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + \sqrt{9} + \sqrt{3} \sqrt{15}}{- 12}$

Step 4.1.3

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

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + \sqrt{3^{2}} + \sqrt{3} \sqrt{15}}{- 12}$

Step 4.1.4

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

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + 3 + \sqrt{3} \sqrt{15}}{- 12}$

Step 4.1.5

Combine using the product rule for radicals.

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + 3 + \sqrt{3 \cdot 15}}{- 12}$

Step 4.1.6

Multiply $3$ by $15$ .

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + 3 + \sqrt{45}}{- 12}$

Step 4.1.7

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

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Step 4.1.7.1

Factor $9$ out of $45$ .

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + 3 + \sqrt{9 (5)}}{- 12}$

Step 4.1.7.2

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

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + 3 + \sqrt{3^{2} \cdot 5}}{- 12}$

Step 4.1.8

Pull terms out from under the radical.

$\frac{- 2 \sqrt{3} - 2 \sqrt{15} + 3 + 3 \sqrt{5}}{- 12}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Move the negative in front of the fraction.

$- \frac{- 2 \sqrt{3} - 2 \sqrt{15} + 3 + 3 \sqrt{5}}{12}$

Step 4.2.2

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

$- \frac{- (2 \sqrt{3}) - 2 \sqrt{15} + 3 + 3 \sqrt{5}}{12}$

Step 4.2.3

Factor $- 1$ out of $- 2 \sqrt{15}$ .

$- \frac{- (2 \sqrt{3}) - (2 \sqrt{15}) + 3 + 3 \sqrt{5}}{12}$

Step 4.2.4

Factor $- 1$ out of $- (2 \sqrt{3}) - (2 \sqrt{15})$ .

$- \frac{- (2 \sqrt{3} + 2 \sqrt{15}) + 3 + 3 \sqrt{5}}{12}$

Step 4.2.5

Rewrite $3$ as $- 1 (- 3)$ .

$- \frac{- (2 \sqrt{3} + 2 \sqrt{15}) - 1 (- 3) + 3 \sqrt{5}}{12}$

Step 4.2.6

Factor $- 1$ out of $- (2 \sqrt{3} + 2 \sqrt{15}) - 1 (- 3)$ .

$- \frac{- (2 \sqrt{3} + 2 \sqrt{15} - 3) + 3 \sqrt{5}}{12}$

Step 4.2.7

Factor $- 1$ out of $3 \sqrt{5}$ .

$- \frac{- (2 \sqrt{3} + 2 \sqrt{15} - 3) - (- 3 \sqrt{5})}{12}$

Step 4.2.8

Factor $- 1$ out of $- (2 \sqrt{3} + 2 \sqrt{15} - 3) - (- 3 \sqrt{5})$ .

$- \frac{- (2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5})}{12}$

Step 4.2.9

Simplify the expression.

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Step 4.2.9.1

Rewrite $- (2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5})$ as $- 1 (2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5})$ .

$- \frac{- 1 (2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5})}{12}$

Step 4.2.9.2

Move the negative in front of the fraction.

$- - \frac{2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5}}{12}$

Step 4.2.9.3

Multiply $- 1$ by $- 1$ .

$1 \frac{2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5}}{12}$

Step 4.2.9.4

Multiply $\frac{2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5}}{12}$ by $1$ .

$\frac{2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5}}{12}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3} + 2 \sqrt{15} - 3 - 3 \sqrt{5}}{12}$

Decimal Form:

$0.12515536 \dots$