Simplify (-2-6i)/(-3+5i)

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

Step 1

Multiply the numerator and denominator of $\frac{- 2 - 6 i}{- 3 + 5 i}$ by the conjugate of $- 3 + 5 i$ to make the denominator real.

$\frac{- 2 - 6 i}{- 3 + 5 i} \cdot \frac{- 3 - 5 i}{- 3 - 5 i}$

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(- 2 - 6 i) (- 3 - 5 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $- 3$ .

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

Multiply $- 5$ by $- 2$ .

$\frac{6 + 10 i - 6 i \cdot - 3 - 6 i (- 5 i)}{(- 3 + 5 i) (- 3 - 5 i)}$

Multiply $- 3$ by $- 6$ .

$\frac{6 + 10 i + 18 i - 6 i (- 5 i)}{(- 3 + 5 i) (- 3 - 5 i)}$

Multiply $- 6 i (- 5 i)$ .

Multiply $- 5$ by $- 6$ .

$\frac{6 + 10 i + 18 i + 30 i i}{(- 3 + 5 i) (- 3 - 5 i)}$

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

$\frac{6 + 10 i + 18 i + 30 (i^{1} i)}{(- 3 + 5 i) (- 3 - 5 i)}$

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

$\frac{6 + 10 i + 18 i + 30 (i^{1} i^{1})}{(- 3 + 5 i) (- 3 - 5 i)}$

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

$\frac{6 + 10 i + 18 i + 30 i^{1 + 1}}{(- 3 + 5 i) (- 3 - 5 i)}$

Add $1$ and $1$ .

$\frac{6 + 10 i + 18 i + 30 i^{2}}{(- 3 + 5 i) (- 3 - 5 i)}$

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

$\frac{6 + 10 i + 18 i + 30 \cdot - 1}{(- 3 + 5 i) (- 3 - 5 i)}$

Multiply $30$ by $- 1$ .

$\frac{6 + 10 i + 18 i - 30}{(- 3 + 5 i) (- 3 - 5 i)}$

Subtract $30$ from $6$ .

$\frac{- 24 + 10 i + 18 i}{(- 3 + 5 i) (- 3 - 5 i)}$

Add $10 i$ and $18 i$ .

$\frac{- 24 + 28 i}{(- 3 + 5 i) (- 3 - 5 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 24 + 28 i}{- 3 (- 3 - 5 i) + 5 i (- 3 - 5 i)}$

Apply the distributive property.

$\frac{- 24 + 28 i}{- 3 \cdot - 3 - 3 (- 5 i) + 5 i (- 3 - 5 i)}$

Apply the distributive property.

$\frac{- 24 + 28 i}{- 3 \cdot - 3 - 3 (- 5 i) + 5 i \cdot - 3 + 5 i (- 5 i)}$

Simplify.

Multiply $- 3$ by $- 3$ .

$\frac{- 24 + 28 i}{9 - 3 (- 5 i) + 5 i \cdot - 3 + 5 i (- 5 i)}$

Multiply $- 5$ by $- 3$ .

$\frac{- 24 + 28 i}{9 + 15 i + 5 i \cdot - 3 + 5 i (- 5 i)}$

Multiply $- 3$ by $5$ .

$\frac{- 24 + 28 i}{9 + 15 i - 15 i + 5 i (- 5 i)}$

Multiply $- 5$ by $5$ .

$\frac{- 24 + 28 i}{9 + 15 i - 15 i - 25 i i}$

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

$\frac{- 24 + 28 i}{9 + 15 i - 15 i - 25 (i^{1} i)}$

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

$\frac{- 24 + 28 i}{9 + 15 i - 15 i - 25 (i^{1} i^{1})}$

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

$\frac{- 24 + 28 i}{9 + 15 i - 15 i - 25 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 24 + 28 i}{9 + 15 i - 15 i - 25 i^{2}}$

Subtract $15 i$ from $15 i$ .

$\frac{- 24 + 28 i}{9 + 0 - 25 i^{2}}$

Add $9$ and $0$ .

$\frac{- 24 + 28 i}{9 - 25 i^{2}}$

Simplify each term.

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

$\frac{- 24 + 28 i}{9 - 25 \cdot - 1}$

Multiply $- 25$ by $- 1$ .

$\frac{- 24 + 28 i}{9 + 25}$

Add $9$ and $25$ .

$\frac{- 24 + 28 i}{34}$

Step 3

Cancel the common factor of $- 24 + 28 i$ and $34$ .

Factor $2$ out of $- 24$ .

$\frac{2 (- 12) + 28 i}{34}$

Factor $2$ out of $28 i$ .

$\frac{2 (- 12) + 2 (14 i)}{34}$

Factor $2$ out of $2 (- 12) + 2 (14 i)$ .

$\frac{2 (- 12 + 14 i)}{34}$

Cancel the common factors.

Factor $2$ out of $34$ .

$\frac{2 (- 12 + 14 i)}{2 \cdot 17}$

Cancel the common factor.

$\frac{2 (- 12 + 14 i)}{2 \cdot 17}$

Rewrite the expression.

$\frac{- 12 + 14 i}{17}$

Step 4

Split the fraction $\frac{- 12 + 14 i}{17}$ into two fractions.

$\frac{- 12}{17} + \frac{14 i}{17}$

Step 5

Move the negative in front of the fraction.

$- \frac{12}{17} + \frac{14 i}{17}$

Simplify (-2-6i)/(-3+5i)

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