Simplify (-10+6i)/(-6+8i)

$\frac{- 10 + 6 i}{- 6 + 8 i}$

Step 1

Cancel the common factor of $- 10 + 6 i$ and $- 6 + 8 i$ .

Factor $2$ out of $- 10$ .

$\frac{2 (- 5) + 6 i}{- 6 + 8 i}$

Factor $2$ out of $6 i$ .

$\frac{2 (- 5) + 2 (3 i)}{- 6 + 8 i}$

Factor $2$ out of $2 (- 5) + 2 (3 i)$ .

$\frac{2 (- 5 + 3 i)}{- 6 + 8 i}$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$\frac{2 (- 5 + 3 i)}{2 \cdot - 3 + 8 i}$

Factor $2$ out of $8 i$ .

$\frac{2 (- 5 + 3 i)}{2 \cdot - 3 + 2 (4 i)}$

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

$\frac{2 (- 5 + 3 i)}{2 (- 3 + 4 i)}$

Cancel the common factor.

$\frac{2 (- 5 + 3 i)}{2 (- 3 + 4 i)}$

Rewrite the expression.

$\frac{- 5 + 3 i}{- 3 + 4 i}$

Step 2

Multiply the numerator and denominator of $\frac{- 5 + 3 i}{- 3 + 4 i}$ by the conjugate of $- 3 + 4 i$ to make the denominator real.

$\frac{- 5 + 3 i}{- 3 + 4 i} \cdot \frac{- 3 - 4 i}{- 3 - 4 i}$

Step 3

Multiply.

Combine.

$\frac{(- 5 + 3 i) (- 3 - 4 i)}{(- 3 + 4 i) (- 3 - 4 i)}$

Simplify the numerator.

Expand $(- 5 + 3 i) (- 3 - 4 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 5 (- 3 - 4 i) + 3 i (- 3 - 4 i)}{(- 3 + 4 i) (- 3 - 4 i)}$

Apply the distributive property.

$\frac{- 5 \cdot - 3 - 5 (- 4 i) + 3 i (- 3 - 4 i)}{(- 3 + 4 i) (- 3 - 4 i)}$

Apply the distributive property.

$\frac{- 5 \cdot - 3 - 5 (- 4 i) + 3 i \cdot - 3 + 3 i (- 4 i)}{(- 3 + 4 i) (- 3 - 4 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 5$ by $- 3$ .

$\frac{15 - 5 (- 4 i) + 3 i \cdot - 3 + 3 i (- 4 i)}{(- 3 + 4 i) (- 3 - 4 i)}$

Multiply $- 4$ by $- 5$ .

$\frac{15 + 20 i + 3 i \cdot - 3 + 3 i (- 4 i)}{(- 3 + 4 i) (- 3 - 4 i)}$

Multiply $- 3$ by $3$ .

$\frac{15 + 20 i - 9 i + 3 i (- 4 i)}{(- 3 + 4 i) (- 3 - 4 i)}$

Multiply $3 i (- 4 i)$ .

Multiply $- 4$ by $3$ .

$\frac{15 + 20 i - 9 i - 12 i i}{(- 3 + 4 i) (- 3 - 4 i)}$

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

$\frac{15 + 20 i - 9 i - 12 (i^{1} i)}{(- 3 + 4 i) (- 3 - 4 i)}$

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

$\frac{15 + 20 i - 9 i - 12 (i^{1} i^{1})}{(- 3 + 4 i) (- 3 - 4 i)}$

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

$\frac{15 + 20 i - 9 i - 12 i^{1 + 1}}{(- 3 + 4 i) (- 3 - 4 i)}$

Add $1$ and $1$ .

$\frac{15 + 20 i - 9 i - 12 i^{2}}{(- 3 + 4 i) (- 3 - 4 i)}$

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

$\frac{15 + 20 i - 9 i - 12 \cdot - 1}{(- 3 + 4 i) (- 3 - 4 i)}$

Multiply $- 12$ by $- 1$ .

$\frac{15 + 20 i - 9 i + 12}{(- 3 + 4 i) (- 3 - 4 i)}$

Add $15$ and $12$ .

$\frac{27 + 20 i - 9 i}{(- 3 + 4 i) (- 3 - 4 i)}$

Subtract $9 i$ from $20 i$ .

$\frac{27 + 11 i}{(- 3 + 4 i) (- 3 - 4 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{27 + 11 i}{- 3 (- 3 - 4 i) + 4 i (- 3 - 4 i)}$

Apply the distributive property.

$\frac{27 + 11 i}{- 3 \cdot - 3 - 3 (- 4 i) + 4 i (- 3 - 4 i)}$

Apply the distributive property.

$\frac{27 + 11 i}{- 3 \cdot - 3 - 3 (- 4 i) + 4 i \cdot - 3 + 4 i (- 4 i)}$

Simplify.

Multiply $- 3$ by $- 3$ .

$\frac{27 + 11 i}{9 - 3 (- 4 i) + 4 i \cdot - 3 + 4 i (- 4 i)}$

Multiply $- 4$ by $- 3$ .

$\frac{27 + 11 i}{9 + 12 i + 4 i \cdot - 3 + 4 i (- 4 i)}$

Multiply $- 3$ by $4$ .

$\frac{27 + 11 i}{9 + 12 i - 12 i + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

$\frac{27 + 11 i}{9 + 12 i - 12 i - 16 i i}$

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

$\frac{27 + 11 i}{9 + 12 i - 12 i - 16 (i^{1} i)}$

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

$\frac{27 + 11 i}{9 + 12 i - 12 i - 16 (i^{1} i^{1})}$

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

$\frac{27 + 11 i}{9 + 12 i - 12 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{27 + 11 i}{9 + 12 i - 12 i - 16 i^{2}}$

Subtract $12 i$ from $12 i$ .

$\frac{27 + 11 i}{9 + 0 - 16 i^{2}}$

Add $9$ and $0$ .

$\frac{27 + 11 i}{9 - 16 i^{2}}$

Simplify each term.

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

$\frac{27 + 11 i}{9 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{27 + 11 i}{9 + 16}$

Add $9$ and $16$ .

$\frac{27 + 11 i}{25}$

Step 4

Split the fraction $\frac{27 + 11 i}{25}$ into two fractions.

$\frac{27}{25} + \frac{11 i}{25}$

Simplify (-10+6i)/(-6+8i)

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