Simplify (-8+6i)/(2-8i)

$\frac{- 8 + 6 i}{2 - 8 i}$

Step 1

Cancel the common factor of $- 8 + 6 i$ and $2 - 8 i$ .

Factor $2$ out of $- 8$ .

$\frac{2 \cdot - 4 + 6 i}{2 - 8 i}$

Factor $2$ out of $6 i$ .

$\frac{2 \cdot - 4 + 2 (3 i)}{2 - 8 i}$

Factor $2$ out of $2 (- 4) + 2 (3 i)$ .

$\frac{2 (- 4 + 3 i)}{2 - 8 i}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 (- 4 + 3 i)}{2 (1) - 8 i}$

Factor $2$ out of $- 8 i$ .

$\frac{2 (- 4 + 3 i)}{2 (1) + 2 (- 4 i)}$

Factor $2$ out of $2 (1) + 2 (- 4 i)$ .

$\frac{2 (- 4 + 3 i)}{2 (1 - 4 i)}$

Cancel the common factor.

$\frac{2 (- 4 + 3 i)}{2 (1 - 4 i)}$

Rewrite the expression.

$\frac{- 4 + 3 i}{1 - 4 i}$

Step 2

Multiply the numerator and denominator of $\frac{- 4 + 3 i}{1 - 4 i}$ by the conjugate of $1 - 4 i$ to make the denominator real.

$\frac{- 4 + 3 i}{1 - 4 i} \cdot \frac{1 + 4 i}{1 + 4 i}$

Step 3

Multiply.

Combine.

$\frac{(- 4 + 3 i) (1 + 4 i)}{(1 - 4 i) (1 + 4 i)}$

Simplify the numerator.

Expand $(- 4 + 3 i) (1 + 4 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 4 (1 + 4 i) + 3 i (1 + 4 i)}{(1 - 4 i) (1 + 4 i)}$

Apply the distributive property.

$\frac{- 4 \cdot 1 - 4 (4 i) + 3 i (1 + 4 i)}{(1 - 4 i) (1 + 4 i)}$

Apply the distributive property.

$\frac{- 4 \cdot 1 - 4 (4 i) + 3 i \cdot 1 + 3 i (4 i)}{(1 - 4 i) (1 + 4 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 4$ by $1$ .

$\frac{- 4 - 4 (4 i) + 3 i \cdot 1 + 3 i (4 i)}{(1 - 4 i) (1 + 4 i)}$

Multiply $4$ by $- 4$ .

$\frac{- 4 - 16 i + 3 i \cdot 1 + 3 i (4 i)}{(1 - 4 i) (1 + 4 i)}$

Multiply $3$ by $1$ .

$\frac{- 4 - 16 i + 3 i + 3 i (4 i)}{(1 - 4 i) (1 + 4 i)}$

Multiply $3 i (4 i)$ .

Multiply $4$ by $3$ .

$\frac{- 4 - 16 i + 3 i + 12 i i}{(1 - 4 i) (1 + 4 i)}$

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

$\frac{- 4 - 16 i + 3 i + 12 (i^{1} i)}{(1 - 4 i) (1 + 4 i)}$

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

$\frac{- 4 - 16 i + 3 i + 12 (i^{1} i^{1})}{(1 - 4 i) (1 + 4 i)}$

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

$\frac{- 4 - 16 i + 3 i + 12 i^{1 + 1}}{(1 - 4 i) (1 + 4 i)}$

Add $1$ and $1$ .

$\frac{- 4 - 16 i + 3 i + 12 i^{2}}{(1 - 4 i) (1 + 4 i)}$

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

$\frac{- 4 - 16 i + 3 i + 12 \cdot - 1}{(1 - 4 i) (1 + 4 i)}$

Multiply $12$ by $- 1$ .

$\frac{- 4 - 16 i + 3 i - 12}{(1 - 4 i) (1 + 4 i)}$

Subtract $12$ from $- 4$ .

$\frac{- 16 - 16 i + 3 i}{(1 - 4 i) (1 + 4 i)}$

Add $- 16 i$ and $3 i$ .

$\frac{- 16 - 13 i}{(1 - 4 i) (1 + 4 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 16 - 13 i}{1 (1 + 4 i) - 4 i (1 + 4 i)}$

Apply the distributive property.

$\frac{- 16 - 13 i}{1 \cdot 1 + 1 (4 i) - 4 i (1 + 4 i)}$

Apply the distributive property.

$\frac{- 16 - 13 i}{1 \cdot 1 + 1 (4 i) - 4 i \cdot 1 - 4 i (4 i)}$

Simplify.

Multiply $1$ by $1$ .

$\frac{- 16 - 13 i}{1 + 1 (4 i) - 4 i \cdot 1 - 4 i (4 i)}$

Multiply $4$ by $1$ .

$\frac{- 16 - 13 i}{1 + 4 i - 4 i \cdot 1 - 4 i (4 i)}$

Multiply $- 4$ by $1$ .

$\frac{- 16 - 13 i}{1 + 4 i - 4 i - 4 i (4 i)}$

Multiply $4$ by $- 4$ .

$\frac{- 16 - 13 i}{1 + 4 i - 4 i - 16 i i}$

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

$\frac{- 16 - 13 i}{1 + 4 i - 4 i - 16 (i^{1} i)}$

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

$\frac{- 16 - 13 i}{1 + 4 i - 4 i - 16 (i^{1} i^{1})}$

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

$\frac{- 16 - 13 i}{1 + 4 i - 4 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 16 - 13 i}{1 + 4 i - 4 i - 16 i^{2}}$

Subtract $4 i$ from $4 i$ .

$\frac{- 16 - 13 i}{1 + 0 - 16 i^{2}}$

Add $1$ and $0$ .

$\frac{- 16 - 13 i}{1 - 16 i^{2}}$

Simplify each term.

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

$\frac{- 16 - 13 i}{1 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{- 16 - 13 i}{1 + 16}$

Add $1$ and $16$ .

$\frac{- 16 - 13 i}{17}$

Step 4

Split the fraction $\frac{- 16 - 13 i}{17}$ into two fractions.

$\frac{- 16}{17} + \frac{- 13 i}{17}$

Step 5

Simplify each term.

Move the negative in front of the fraction.

$- \frac{16}{17} + \frac{- 13 i}{17}$

Move the negative in front of the fraction.

$- \frac{16}{17} - \frac{13 i}{17}$

Simplify (-8+6i)/(2-8i)

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