Algebra Examples

$\frac{3 + \sqrt{8}}{2 - 2 \sqrt{8}}$

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 $8$ .

$\frac{3 + \sqrt{4 (2)}}{2 - 2 \sqrt{8}}$

Step 1.1.2

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

$\frac{3 + \sqrt{2^{2} \cdot 2}}{2 - 2 \sqrt{8}}$

Step 1.2

Pull terms out from under the radical.

$\frac{3 + | 2 | \sqrt{2}}{2 - 2 \sqrt{8}}$

Step 1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{3 + 2 \sqrt{2}}{2 - 2 \sqrt{8}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $8$ .

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

Step 2.1.2

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

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

Step 2.2

Pull terms out from under the radical.

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

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 2.4

Multiply $2$ by $- 2$ .

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

Step 3

Multiply $\frac{3 + 2 \sqrt{2}}{2 - 4 \sqrt{2}}$ by $\frac{2 + 4 \sqrt{2}}{2 + 4 \sqrt{2}}$ .

$\frac{3 + 2 \sqrt{2}}{2 - 4 \sqrt{2}} \cdot \frac{2 + 4 \sqrt{2}}{2 + 4 \sqrt{2}}$

Step 4

Multiply $\frac{3 + 2 \sqrt{2}}{2 - 4 \sqrt{2}}$ by $\frac{2 + 4 \sqrt{2}}{2 + 4 \sqrt{2}}$ .

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

Step 5

Expand the denominator using the FOIL method.

$\frac{(3 + 2 \sqrt{2}) (2 + 4 \sqrt{2})}{4 + 8 \sqrt{2} - 8 \sqrt{2} - 16 {\sqrt{2}}^{2}}$

Step 6

Simplify.

$\frac{(3 + 2 \sqrt{2}) (2 + 4 \sqrt{2})}{- 28}$

Step 7

Cancel the common factor of $2 + 4 \sqrt{2}$ and $- 28$ .

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Step 7.1

Factor $2$ out of $(3 + 2 \sqrt{2}) (2 + 4 \sqrt{2})$ .

$\frac{2 ((3 + 2 \sqrt{2}) (1 + 2 \sqrt{2}))}{- 28}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $2$ out of $- 28$ .

$\frac{2 ((3 + 2 \sqrt{2}) (1 + 2 \sqrt{2}))}{2 (- 14)}$

Step 7.2.2

Cancel the common factor.

$\frac{2 ((3 + 2 \sqrt{2}) (1 + 2 \sqrt{2}))}{2 \cdot - 14}$

Step 7.2.3

Rewrite the expression.

$\frac{(3 + 2 \sqrt{2}) (1 + 2 \sqrt{2})}{- 14}$

Step 8

Expand $(3 + 2 \sqrt{2}) (1 + 2 \sqrt{2})$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$\frac{3 (1 + 2 \sqrt{2}) + 2 \sqrt{2} (1 + 2 \sqrt{2})}{- 14}$

Step 8.2

Apply the distributive property.

$\frac{3 \cdot 1 + 3 (2 \sqrt{2}) + 2 \sqrt{2} (1 + 2 \sqrt{2})}{- 14}$

Step 8.3

Apply the distributive property.

$\frac{3 \cdot 1 + 3 (2 \sqrt{2}) + 2 \sqrt{2} \cdot 1 + 2 \sqrt{2} (2 \sqrt{2})}{- 14}$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $3$ by $1$ .

$\frac{3 + 3 (2 \sqrt{2}) + 2 \sqrt{2} \cdot 1 + 2 \sqrt{2} (2 \sqrt{2})}{- 14}$

Step 9.1.2

Multiply $2$ by $3$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} \cdot 1 + 2 \sqrt{2} (2 \sqrt{2})}{- 14}$

Step 9.1.3

Multiply $2$ by $1$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 2 \sqrt{2} (2 \sqrt{2})}{- 14}$

Step 9.1.4

Multiply $2 \sqrt{2} (2 \sqrt{2})$ .

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Step 9.1.4.1

Multiply $2$ by $2$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 \sqrt{2} \sqrt{2}}{- 14}$

Step 9.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 ({\sqrt{2}}^{1} \sqrt{2})}{- 14}$

Step 9.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{- 14}$

Step 9.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1}}{- 14}$

Step 9.1.4.5

Add $1$ and $1$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 {\sqrt{2}}^{2}}{- 14}$

Step 9.1.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 9.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2}}{- 14}$

Step 9.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2}}{- 14}$

Step 9.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}}{- 14}$

Step 9.1.5.4

Cancel the common factor of $2$ .

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Step 9.1.5.4.1

Cancel the common factor.

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}}{- 14}$

Step 9.1.5.4.2

Rewrite the expression.

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 \cdot 2^{1}}{- 14}$

Step 9.1.5.5

Evaluate the exponent.

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 4 \cdot 2}{- 14}$

Step 9.1.6

Multiply $4$ by $2$ .

$\frac{3 + 6 \sqrt{2} + 2 \sqrt{2} + 8}{- 14}$

Step 9.2

Add $3$ and $8$ .

$\frac{11 + 6 \sqrt{2} + 2 \sqrt{2}}{- 14}$

Step 9.3

Add $6 \sqrt{2}$ and $2 \sqrt{2}$ .

$\frac{11 + 8 \sqrt{2}}{- 14}$

Step 10

Move the negative in front of the fraction.

$- \frac{11 + 8 \sqrt{2}}{14}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{11 + 8 \sqrt{2}}{14}$

Decimal Form:

$- 1.59383632 \dots$