Algebra Examples

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

Step 1

Multiply the numerator and denominator of $\frac{- 1 - i}{4 + 4 i}$ by the conjugate of $4 + 4 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(- 1 - i) (4 - 4 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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 $- 1$ by $4$ .

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

Step 2.2.2.1.2

Multiply $- 4$ by $- 1$ .

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

Step 2.2.2.1.3

Multiply $4$ by $- 1$ .

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

Step 2.2.2.1.4

Multiply $- i (- 4 i)$ .

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Step 2.2.2.1.4.1

Multiply $- 4$ by $- 1$ .

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

Step 2.2.2.1.4.2

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

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

Step 2.2.2.1.4.3

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

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

Step 2.2.2.1.4.4

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

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

Step 2.2.2.1.4.5

Add $1$ and $1$ .

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

Step 2.2.2.1.5

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

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

Step 2.2.2.1.6

Multiply $4$ by $- 1$ .

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

Step 2.2.2.2

Subtract $4$ from $- 4$ .

$\frac{- 8 + 4 i - 4 i}{(4 + 4 i) (4 - 4 i)}$

Step 2.2.2.3

Subtract $4 i$ from $4 i$ .

$\frac{- 8 + 0}{(4 + 4 i) (4 - 4 i)}$

Step 2.2.2.4

Add $- 8$ and $0$ .

$\frac{- 8}{(4 + 4 i) (4 - 4 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(4 + 4 i) (4 - 4 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 8}{4 (4 - 4 i) + 4 i (4 - 4 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 8}{4 \cdot 4 + 4 (- 4 i) + 4 i (4 - 4 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 8}{4 \cdot 4 + 4 (- 4 i) + 4 i \cdot 4 + 4 i (- 4 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{- 8}{16 + 4 (- 4 i) + 4 i \cdot 4 + 4 i (- 4 i)}$

Step 2.3.2.2

Multiply $- 4$ by $4$ .

$\frac{- 8}{16 - 16 i + 4 i \cdot 4 + 4 i (- 4 i)}$

Step 2.3.2.3

Multiply $4$ by $4$ .

$\frac{- 8}{16 - 16 i + 16 i + 4 i (- 4 i)}$

Step 2.3.2.4

Multiply $- 4$ by $4$ .

$\frac{- 8}{16 - 16 i + 16 i - 16 i i}$

Step 2.3.2.5

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

$\frac{- 8}{16 - 16 i + 16 i - 16 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 8}{16 - 16 i + 16 i - 16 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 8}{16 - 16 i + 16 i - 16 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 8}{16 - 16 i + 16 i - 16 i^{2}}$

Step 2.3.2.9

Add $- 16 i$ and $16 i$ .

$\frac{- 8}{16 + 0 - 16 i^{2}}$

Step 2.3.2.10

Add $16$ and $0$ .

$\frac{- 8}{16 - 16 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 8}{16 - 16 \cdot - 1}$

Step 2.3.3.2

Multiply $- 16$ by $- 1$ .

$\frac{- 8}{16 + 16}$

Step 2.3.4

Add $16$ and $16$ .

$\frac{- 8}{32}$

Step 3

Cancel the common factor of $- 8$ and $32$ .

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Step 3.1

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{32}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $8$ out of $32$ .

$\frac{8 \cdot - 1}{8 \cdot 4}$

Step 3.2.2

Cancel the common factor.

$\frac{8 \cdot - 1}{8 \cdot 4}$

Step 3.2.3

Rewrite the expression.

$\frac{- 1}{4}$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$