Algebra Examples

$\frac{\sqrt{11}}{2 - \sqrt{5}}$

Step 1

Multiply $\frac{\sqrt{11}}{2 - \sqrt{5}}$ by $\frac{2 + \sqrt{5}}{2 + \sqrt{5}}$ .

$\frac{\sqrt{11}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{\sqrt{11}}{2 - \sqrt{5}}$ by $\frac{2 + \sqrt{5}}{2 + \sqrt{5}}$ .

$\frac{\sqrt{11} (2 + \sqrt{5})}{(2 - \sqrt{5}) (2 + \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{\sqrt{11} (2 + \sqrt{5})}{4 + 2 \sqrt{5} - 2 \sqrt{5} - {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{\sqrt{11} (2 + \sqrt{5})}{- 1}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move the negative one from the denominator of $\frac{\sqrt{11} (2 + \sqrt{5})}{- 1}$ .

$- 1 \cdot (\sqrt{11} (2 + \sqrt{5}))$

Step 2.4.2

Rewrite $- 1 \cdot (\sqrt{11} (2 + \sqrt{5}))$ as $- (\sqrt{11} (2 + \sqrt{5}))$ .

$- (\sqrt{11} (2 + \sqrt{5}))$

Step 2.5

Apply the distributive property.

$- (\sqrt{11} \cdot 2 + \sqrt{11} \sqrt{5})$

Step 2.6

Move $2$ to the left of $\sqrt{11}$ .

$- (2 \cdot \sqrt{11} + \sqrt{11} \sqrt{5})$

Step 2.7

Combine using the product rule for radicals.

$- (2 \cdot \sqrt{11} + \sqrt{11 \cdot 5})$

Step 3

Multiply $11$ by $5$ .

$- (2 \sqrt{11} + \sqrt{55})$

Step 4

Simplify by multiplying through.

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Step 4.1

Apply the distributive property.

$- (2 \sqrt{11}) - \sqrt{55}$

Step 4.2

Multiply $2$ by $- 1$ .

$- 2 \sqrt{11} - \sqrt{55}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- 2 \sqrt{11} - \sqrt{55}$

Decimal Form:

$- 14.04944806 \dots$