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

Multiply the numerator and denominator of $\frac{2 + i \cdot \sqrt{8}}{1 + 1.41421356 \cdot i}$ by the conjugate of $1 + 1.41421356 \cdot i$ to make the denominator real.

$\frac{2 + i \cdot \sqrt{8}}{1 + 1.41421356 \cdot i} \cdot \frac{1 - 1.41421356 i}{1 - 1.41421356 i}$

Step 3

Multiply.

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Step 3.1

Combine.

$\frac{(2 + i \cdot \sqrt{8}) (1 - 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Factor $4$ out of $8$ .

$\frac{(2 + i \cdot \sqrt{4 (2)}) (1 - 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.1.1.2

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

$\frac{(2 + i \cdot \sqrt{2^{2} \cdot 2}) (1 - 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.1.2

Pull terms out from under the radical.

$\frac{(2 + i \cdot (2 \sqrt{2})) (1 - 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.1.3

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

$\frac{(2 + 2 i \sqrt{2}) (1 - 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.2

Expand $(2 + 2 i \sqrt{2}) (1 - 1.41421356 i)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$\frac{2 (1 - 1.41421356 i) + 2 i \sqrt{2} (1 - 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.2.2

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- 1.41421356 i) + 2 i \sqrt{2} (1 - 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.2.3

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- 1.41421356 i) + 2 i \sqrt{2} \cdot 1 + 2 i \sqrt{2} (- 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply $2$ by $1$ .

$\frac{2 + 2 (- 1.41421356 i) + 2 i \sqrt{2} \cdot 1 + 2 i \sqrt{2} (- 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.2

Multiply $- 1.41421356$ by $2$ .

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} \cdot 1 + 2 i \sqrt{2} (- 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.3

Multiply $2$ by $1$ .

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} + 2 i \sqrt{2} (- 1.41421356 i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.4

Multiply $2 i \sqrt{2} (- 1.41421356 i)$ .

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Step 3.2.3.1.4.1

Multiply $- 1.41421356$ by $2$ .

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} - 2.82842712 i \sqrt{2} i}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.4.2

Multiply $\sqrt{2}$ by $- 2.82842712$ .

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} - 4 i i}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.4.3

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

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} - 4 (i^{1} i)}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.4.4

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

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} - 4 (i^{1} i^{1})}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.4.5

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

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} - 4 i^{1 + 1}}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.4.6

Add $1$ and $1$ .

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} - 4 i^{2}}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.5

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

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} - 4 \cdot - 1}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.1.6

Multiply $- 4$ by $- 1$ .

$\frac{2 - 2.82842712 i + 2 i \sqrt{2} + 4}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.2.3.2

Add $2$ and $4$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand $(1 + 1.41421356 \cdot i) (1 - 1.41421356 i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 (1 - 1.41421356 i) + 1.41421356 \cdot i (1 - 1.41421356 i)}$

Step 3.3.1.2

Apply the distributive property.

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 \cdot 1 + 1 (- 1.41421356 i) + 1.41421356 \cdot i (1 - 1.41421356 i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 \cdot 1 + 1 (- 1.41421356 i) + 1.41421356 \cdot i \cdot 1 + 1.41421356 \cdot i (- 1.41421356 i)}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $1$ by $1$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 + 1 (- 1.41421356 i) + 1.41421356 \cdot i \cdot 1 + 1.41421356 \cdot i (- 1.41421356 i)}$

Step 3.3.2.2

Multiply $- 1.41421356$ by $1$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 - 1.41421356 i + 1.41421356 \cdot i \cdot 1 + 1.41421356 \cdot i (- 1.41421356 i)}$

Step 3.3.2.3

Multiply $1.41421356$ by $1$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 - 1.41421356 i + 1.41421356 \cdot i + 1.41421356 \cdot i (- 1.41421356 i)}$

Step 3.3.2.4

Multiply $- 1.41421356$ by $1.41421356$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 - 1.41421356 i + 1.41421356 \cdot i - 2 \cdot i i}$

Step 3.3.2.5

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

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 - 1.41421356 i + 1.41421356 \cdot i - 2 \cdot (i^{1} i)}$

Step 3.3.2.6

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

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 - 1.41421356 i + 1.41421356 \cdot i - 2 \cdot (i^{1} i^{1})}$

Step 3.3.2.7

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

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 - 1.41421356 i + 1.41421356 \cdot i - 2 \cdot i^{1 + 1}}$

Step 3.3.2.8

Add $1$ and $1$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 - 1.41421356 i + 1.41421356 \cdot i - 2 \cdot i^{2}}$

Step 3.3.2.9

Add $- 1.41421356 i$ and $1.41421356 \cdot i$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 + 0 i - 2 \cdot i^{2}}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

Multiply $0$ by $i$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 + 0 - 2 \cdot i^{2}}$

Step 3.3.3.2

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

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 + 0 - 2 \cdot - 1}$

Step 3.3.3.3

Multiply $- 2$ by $- 1$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 + 0 + 2}$

Step 3.3.4

Add $1$ and $0$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1 + 2}$

Step 3.3.5

Add $1$ and $2$ .

$\frac{6 - 2.82842712 i + 2 i \sqrt{2}}{3}$

Step 4

Rewrite $6$ as $1 (6)$ .

$\frac{1 (6) - 2.82842712 i + 2 i \sqrt{2}}{3}$

Step 5

Factor $1$ out of $- 2.82842712 i$ .

$\frac{1 (6) + 1 (- 2.82842712 i) + 2 i \sqrt{2}}{3}$

Step 6

Factor $1$ out of $1 (6) + 1 (- 2.82842712 i)$ .

$\frac{1 (6 - 2.82842712 i) + 2 i \sqrt{2}}{3}$

Step 7

Factor $1$ out of $2 i \sqrt{2}$ .

$\frac{1 (6 - 2.82842712 i) + 1 (2 i \sqrt{2})}{3}$

Step 8

Factor $1$ out of $1 (6 - 2.82842712 i) + 1 (2 i \sqrt{2})$ .

$\frac{1 (6 - 2.82842712 i + 2 i \sqrt{2})}{3}$

Step 9

Factor $3$ out of $3$ .

$\frac{1 (6 - 2.82842712 i + 2 i \sqrt{2})}{3 (1)}$

Step 10

Separate fractions.

$\frac{1}{3} \cdot \frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1}$

Step 11

Divide $1$ by $3$ .

$0.33333333 \frac{6 - 2.82842712 i + 2 i \sqrt{2}}{1}$

Step 12

Divide $6 - 2.82842712 i + 2 i \sqrt{2}$ by $1$ .

$0.33333333 (6 - 2.82842712 i + 2 i \sqrt{2})$

Step 13

Apply the distributive property.

$0.33333333 \cdot 6 + 0.33333333 (- 2.82842712 i) + 0.33333333 (2 i \sqrt{2})$

Step 14

Simplify.

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Step 14.1

Multiply $0.33333333$ by $6$ .

$2 + 0.33333333 (- 2.82842712 i) + 0.33333333 (2 i \sqrt{2})$

Step 14.2

Multiply $- 2.82842712$ by $0.33333333$ .

$2 - 0.94280904 i + 0.33333333 (2 i \sqrt{2})$

Step 14.3

Multiply $0.33333333 (2 i \sqrt{2})$ .

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Step 14.3.1

Multiply $2$ by $0.33333333$ .

$2 - 0.94280904 i + 0.66666666 (i \sqrt{2})$

Step 14.3.2

Multiply $\sqrt{2}$ by $0.66666666$ .

$2 - 0.94280904 i + 0.94280904 i$

Step 15

Add $- 0.94280904 i$ and $0.94280904 i$ .

$2 + 0 i$

Step 16

Multiply $0$ by $i$ .

$2 + 0$

Step 17

Add $2$ and $0$ .

$2$