Simplify the Radical Expression (3+ square root of 8)/(2-2 square root of 8)

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Algebra

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

Simplify the numerator.

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

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Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Simplify the denominator.

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

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Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $- 2$ .

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

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

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

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

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

Cancel the common factors.

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Factor $2$ out of $- 28$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $3$ by $1$ .

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $1$ .

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

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

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Multiply $2$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $4$ by $2$ .

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

Add $3$ and $8$ .

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

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

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

Move the negative in front of the fraction.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.59383632 \dots$

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