Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

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

Step 3.3

Simplify.

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.5

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

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Combine using the product rule for radicals.

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

Step 3.8

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

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

Step 4

Simplify each term.

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Step 4.1

Multiply $2$ by $2$ .

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

Step 4.2

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

$- (\sqrt{2^{2}} + 2 \cdot \sqrt{2})$

Step 4.3

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

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

Step 5

Simplify by multiplying through.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply.

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Step 5.2.1

Multiply $- 1$ by $2$ .

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

Step 5.2.2

Multiply $2$ by $- 1$ .

$- 2 - 2 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- 2 - 2 \sqrt{2}$

Decimal Form:

$- 4.82842712 \dots$