Algebra Examples

$\frac{\sqrt{3}}{- 1 - \sqrt{5}}$

Step 1

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

$\frac{\sqrt{3}}{- 1 - \sqrt{5}} \cdot \frac{- 1 + \sqrt{5}}{- 1 + \sqrt{5}}$

Step 2

Simplify terms.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{\sqrt{3} (- 1 + \sqrt{5})}{- 4}$

Step 2.4

Apply the distributive property.

$\frac{\sqrt{3} \cdot - 1 + \sqrt{3} \sqrt{5}}{- 4}$

Step 2.5

Move $- 1$ to the left of $\sqrt{3}$ .

$\frac{- 1 \cdot \sqrt{3} + \sqrt{3} \sqrt{5}}{- 4}$

Step 2.6

Combine using the product rule for radicals.

$\frac{- 1 \cdot \sqrt{3} + \sqrt{3 \cdot 5}}{- 4}$

Step 3

Simplify each term.

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Step 3.1

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$\frac{- \sqrt{3} + \sqrt{3 \cdot 5}}{- 4}$

Step 3.2

Multiply $3$ by $5$ .

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

Step 4

Simplify with factoring out.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

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

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

Step 4.3

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

$- \frac{- (\sqrt{3}) - 1 (- \sqrt{15})}{4}$

Step 4.4

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

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

Step 4.5

Simplify the expression.

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Step 4.5.1

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

$- \frac{- 1 (\sqrt{3} - \sqrt{15})}{4}$

Step 4.5.2

Move the negative in front of the fraction.

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

Step 4.5.3

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{3} - \sqrt{15}}{4}$

Step 4.5.4

Multiply $\frac{\sqrt{3} - \sqrt{15}}{4}$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.53523313 \dots$