Simplify (-4-4i)/(10-2i)

$\frac{- 4 - 4 i}{10 - 2 i}$

Step 1

Cancel the common factor of $- 4 - 4 i$ and $10 - 2 i$ .

Factor $2$ out of $- 4$ .

$\frac{2 (- 2) - 4 i}{10 - 2 i}$

Factor $2$ out of $- 4 i$ .

$\frac{2 (- 2) + 2 (- 2 i)}{10 - 2 i}$

Factor $2$ out of $2 (- 2) + 2 (- 2 i)$ .

$\frac{2 (- 2 - 2 i)}{10 - 2 i}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 (- 2 - 2 i)}{2 \cdot 5 - 2 i}$

Factor $2$ out of $- 2 i$ .

$\frac{2 (- 2 - 2 i)}{2 \cdot 5 + 2 (- i)}$

Factor $2$ out of $2 (5) + 2 (- i)$ .

$\frac{2 (- 2 - 2 i)}{2 (5 - i)}$

Cancel the common factor.

$\frac{2 (- 2 - 2 i)}{2 (5 - i)}$

Rewrite the expression.

$\frac{- 2 - 2 i}{5 - i}$

Step 2

Multiply the numerator and denominator of $\frac{- 2 - 2 i}{5 - i}$ by the conjugate of $5 - i$ to make the denominator real.

$\frac{- 2 - 2 i}{5 - i} \cdot \frac{5 + i}{5 + i}$

Step 3

Multiply.

Combine.

$\frac{(- 2 - 2 i) (5 + i)}{(5 - i) (5 + i)}$

Simplify the numerator.

Expand $(- 2 - 2 i) (5 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 2 (5 + i) - 2 i (5 + i)}{(5 - i) (5 + i)}$

Apply the distributive property.

$\frac{- 2 \cdot 5 - 2 i - 2 i (5 + i)}{(5 - i) (5 + i)}$

Apply the distributive property.

$\frac{- 2 \cdot 5 - 2 i - 2 i \cdot 5 - 2 i i}{(5 - i) (5 + i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $5$ .

$\frac{- 10 - 2 i - 2 i \cdot 5 - 2 i i}{(5 - i) (5 + i)}$

Multiply $5$ by $- 2$ .

$\frac{- 10 - 2 i - 10 i - 2 i i}{(5 - i) (5 + i)}$

Multiply $- 2 i i$ .

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

$\frac{- 10 - 2 i - 10 i - 2 (i^{1} i)}{(5 - i) (5 + i)}$

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

$\frac{- 10 - 2 i - 10 i - 2 (i^{1} i^{1})}{(5 - i) (5 + i)}$

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

$\frac{- 10 - 2 i - 10 i - 2 i^{1 + 1}}{(5 - i) (5 + i)}$

Add $1$ and $1$ .

$\frac{- 10 - 2 i - 10 i - 2 i^{2}}{(5 - i) (5 + i)}$

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

$\frac{- 10 - 2 i - 10 i - 2 \cdot - 1}{(5 - i) (5 + i)}$

Multiply $- 2$ by $- 1$ .

$\frac{- 10 - 2 i - 10 i + 2}{(5 - i) (5 + i)}$

Add $- 10$ and $2$ .

$\frac{- 8 - 2 i - 10 i}{(5 - i) (5 + i)}$

Subtract $10 i$ from $- 2 i$ .

$\frac{- 8 - 12 i}{(5 - i) (5 + i)}$

Simplify the denominator.

Expand $(5 - i) (5 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 8 - 12 i}{5 (5 + i) - i (5 + i)}$

Apply the distributive property.

$\frac{- 8 - 12 i}{5 \cdot 5 + 5 i - i (5 + i)}$

Apply the distributive property.

$\frac{- 8 - 12 i}{5 \cdot 5 + 5 i - i \cdot 5 - i i}$

Simplify.

Multiply $5$ by $5$ .

$\frac{- 8 - 12 i}{25 + 5 i - i \cdot 5 - i i}$

Multiply $5$ by $- 1$ .

$\frac{- 8 - 12 i}{25 + 5 i - 5 i - i i}$

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

$\frac{- 8 - 12 i}{25 + 5 i - 5 i - (i^{1} i)}$

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

$\frac{- 8 - 12 i}{25 + 5 i - 5 i - (i^{1} i^{1})}$

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

$\frac{- 8 - 12 i}{25 + 5 i - 5 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 8 - 12 i}{25 + 5 i - 5 i - i^{2}}$

Subtract $5 i$ from $5 i$ .

$\frac{- 8 - 12 i}{25 + 0 - i^{2}}$

Add $25$ and $0$ .

$\frac{- 8 - 12 i}{25 - i^{2}}$

Simplify each term.

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

$\frac{- 8 - 12 i}{25 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{- 8 - 12 i}{25 + 1}$

Add $25$ and $1$ .

$\frac{- 8 - 12 i}{26}$

Step 4

Cancel the common factor of $- 8 - 12 i$ and $26$ .

Factor $2$ out of $- 8$ .

$\frac{2 (- 4) - 12 i}{26}$

Factor $2$ out of $- 12 i$ .

$\frac{2 (- 4) + 2 (- 6 i)}{26}$

Factor $2$ out of $2 (- 4) + 2 (- 6 i)$ .

$\frac{2 (- 4 - 6 i)}{26}$

Cancel the common factors.

Factor $2$ out of $26$ .

$\frac{2 (- 4 - 6 i)}{2 \cdot 13}$

Cancel the common factor.

$\frac{2 (- 4 - 6 i)}{2 \cdot 13}$

Rewrite the expression.

$\frac{- 4 - 6 i}{13}$

Step 5

Split the fraction $\frac{- 4 - 6 i}{13}$ into two fractions.

$\frac{- 4}{13} + \frac{- 6 i}{13}$

Step 6

Simplify each term.

Move the negative in front of the fraction.

$- \frac{4}{13} + \frac{- 6 i}{13}$

Move the negative in front of the fraction.

$- \frac{4}{13} - \frac{6 i}{13}$

Simplify (-4-4i)/(10-2i)

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