Algebra Examples

$\frac{2 + \sqrt{- 8}}{1 + \sqrt{- 2}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{2 + \sqrt{- 1 (8)}}{1 + \sqrt{- 2}}$

Step 1.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\frac{2 + \sqrt{- 1} \cdot \sqrt{8}}{1 + \sqrt{- 2}}$

Step 1.3

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

$\frac{2 + i \cdot \sqrt{8}}{1 + \sqrt{- 2}}$

Step 1.4

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{2 + i \cdot \sqrt{8}}{1 + \sqrt{- 1 (2)}}$

Step 1.5

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

$\frac{2 + i \cdot \sqrt{8}}{1 + \sqrt{- 1} \cdot \sqrt{2}}$

Step 1.6

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

$\frac{2 + i \cdot \sqrt{8}}{1 + i \cdot \sqrt{2}}$

Step 2

Simplify the numerator.

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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 $8$ .

$\frac{2 + i \cdot \sqrt{4 (2)}}{1 + i \cdot \sqrt{2}}$

Step 2.1.2

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

$\frac{2 + i \cdot \sqrt{2^{2} \cdot 2}}{1 + i \cdot \sqrt{2}}$

Step 2.2

Pull terms out from under the radical.

$\frac{2 + i \cdot (2 \sqrt{2})}{1 + i \cdot \sqrt{2}}$

Step 2.3

Move $2$ to the left of $i$ .

$\frac{2 + 2 i \sqrt{2}}{1 + i \cdot \sqrt{2}}$

Step 3

Multiply $\frac{2 + 2 i \sqrt{2}}{1 + i \cdot \sqrt{2}}$ by $\frac{1 - i \cdot \sqrt{2}}{1 - i \cdot \sqrt{2}}$ .

$\frac{2 + 2 i \sqrt{2}}{1 + i \cdot \sqrt{2}} \cdot \frac{1 - i \cdot \sqrt{2}}{1 - i \cdot \sqrt{2}}$

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{2 + 2 i \sqrt{2}}{1 + i \cdot \sqrt{2}}$ by $\frac{1 - i \cdot \sqrt{2}}{1 - i \cdot \sqrt{2}}$ .

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

Step 4.2

Expand the denominator using the FOIL method.

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

Step 4.3

Simplify.

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $2$ by $1$ .

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

Step 6.1.2

Multiply $- 1$ by $2$ .

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

Step 6.1.3

Multiply $2$ by $1$ .

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

Step 6.1.4

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

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Step 6.1.4.1

Multiply $- 1$ by $2$ .

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

Step 6.1.4.2

Raise $i$ to the power of $1$ .

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

Step 6.1.4.3

Raise $i$ to the power of $1$ .

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

Step 6.1.4.4

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

$\frac{2 - 2 i \sqrt{2} + 2 i \sqrt{2} - 2 i^{1 + 1} \sqrt{2} \sqrt{2}}{3}$

Step 6.1.4.5

Add $1$ and $1$ .

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

Step 6.1.4.6

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

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

Step 6.1.4.7

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

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

Step 6.1.4.8

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

$\frac{2 - 2 i \sqrt{2} + 2 i \sqrt{2} - 2 i^{2} {\sqrt{2}}^{1 + 1}}{3}$

Step 6.1.4.9

Add $1$ and $1$ .

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

Step 6.1.5

Rewrite $i^{2}$ as $- 1$ .

$\frac{2 - 2 i \sqrt{2} + 2 i \sqrt{2} - 2 \cdot - 1 {\sqrt{2}}^{2}}{3}$

Step 6.1.6

Multiply $- 2$ by $- 1$ .

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

Step 6.1.7

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

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Step 6.1.7.1

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

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

Step 6.1.7.2

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

$\frac{2 - 2 i \sqrt{2} + 2 i \sqrt{2} + 2 \cdot 2^{\frac{1}{2} \cdot 2}}{3}$

Step 6.1.7.3

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

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

Step 6.1.7.4

Cancel the common factor of $2$ .

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Step 6.1.7.4.1

Cancel the common factor.

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

Step 6.1.7.4.2

Rewrite the expression.

$\frac{2 - 2 i \sqrt{2} + 2 i \sqrt{2} + 2 \cdot 2^{1}}{3}$

Step 6.1.7.5

Evaluate the exponent.

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

Step 6.1.8

Multiply $2$ by $2$ .

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

Step 6.2

Add $2$ and $4$ .

$\frac{- 2 i \sqrt{2} + 2 i \sqrt{2} + 6}{3}$

Step 6.3

Add $- 2 i \sqrt{2}$ and $2 i \sqrt{2}$ .

$\frac{0 + 6}{3}$

Step 6.4

Add $0$ and $6$ .

$\frac{6}{3}$

Step 7

Divide $6$ by $3$ .

$2$