Algebra Examples

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

Step 1

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(1 - i) (- 2 - 2 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $- 2$ by $1$ .

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

Step 2.2.2.1.2

Multiply $- 2 i$ by $1$ .

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

Step 2.2.2.1.3

Multiply $- 2$ by $- 1$ .

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

Step 2.2.2.1.4

Multiply $- i (- 2 i)$ .

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Step 2.2.2.1.4.1

Multiply $- 2$ by $- 1$ .

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

Step 2.2.2.1.4.2

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

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

Step 2.2.2.1.4.3

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

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

Step 2.2.2.1.4.4

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

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

Step 2.2.2.1.4.5

Add $1$ and $1$ .

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

Step 2.2.2.1.5

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

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

Step 2.2.2.1.6

Multiply $2$ by $- 1$ .

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

Step 2.2.2.2

Subtract $2$ from $- 2$ .

$\frac{- 4 - 2 i + 2 i}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2.3

Add $- 2 i$ and $2 i$ .

$\frac{- 4 + 0}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2.4

Add $- 4$ and $0$ .

$\frac{- 4}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(- 2 + 2 i) (- 2 - 2 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 4}{- 2 (- 2 - 2 i) + 2 i (- 2 - 2 i)}$

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $- 2$ by $- 2$ .

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

Step 2.3.2.2

Multiply $- 2$ by $- 2$ .

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

Step 2.3.2.3

Multiply $- 2$ by $2$ .

$\frac{- 4}{4 + 4 i - 4 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply $- 2$ by $2$ .

$\frac{- 4}{4 + 4 i - 4 i - 4 i i}$

Step 2.3.2.5

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

$\frac{- 4}{4 + 4 i - 4 i - 4 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 4}{4 + 4 i - 4 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 4}{4 + 4 i - 4 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 4}{4 + 4 i - 4 i - 4 i^{2}}$

Step 2.3.2.9

Subtract $4 i$ from $4 i$ .

$\frac{- 4}{4 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add $4$ and $0$ .

$\frac{- 4}{4 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 4}{4 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{- 4}{4 + 4}$

Step 2.3.4

Add $4$ and $4$ .

$\frac{- 4}{8}$

Step 3

Cancel the common factor of $- 4$ and $8$ .

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Step 3.1

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{8}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot - 1}{4 \cdot 2}$

Step 3.2.2

Cancel the common factor.

$\frac{4 \cdot - 1}{4 \cdot 2}$

Step 3.2.3

Rewrite the expression.

$\frac{- 1}{2}$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$