Divide (4+3i)/(-10+i)

$\frac{4 + 3 i}{- 10 + i}$

Step 1

Multiply the numerator and denominator of $\frac{4 + 3 i}{- 10 + 1 i}$ by the conjugate of $- 10 + 1 i$ to make the denominator real.

$\frac{4 + 3 i}{- 10 + 1 i} \cdot \frac{- 10 - i}{- 10 - i}$

Step 2

Multiply.

Combine.

$\frac{(4 + 3 i) (- 10 - i)}{(- 10 + 1 i) (- 10 - i)}$

Simplify the numerator.

Expand $(4 + 3 i) (- 10 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{4 (- 10 - i) + 3 i (- 10 - i)}{(- 10 + 1 i) (- 10 - i)}$

Apply the distributive property.

$\frac{4 \cdot - 10 + 4 (- i) + 3 i (- 10 - i)}{(- 10 + 1 i) (- 10 - i)}$

Apply the distributive property.

$\frac{4 \cdot - 10 + 4 (- i) + 3 i \cdot - 10 + 3 i (- i)}{(- 10 + 1 i) (- 10 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $- 10$ .

$\frac{- 40 + 4 (- i) + 3 i \cdot - 10 + 3 i (- i)}{(- 10 + 1 i) (- 10 - i)}$

Multiply $- 1$ by $4$ .

$\frac{- 40 - 4 i + 3 i \cdot - 10 + 3 i (- i)}{(- 10 + 1 i) (- 10 - i)}$

Multiply $- 10$ by $3$ .

$\frac{- 40 - 4 i - 30 i + 3 i (- i)}{(- 10 + 1 i) (- 10 - i)}$

Multiply $3 i (- i)$ .

Multiply $- 1$ by $3$ .

$\frac{- 40 - 4 i - 30 i - 3 i i}{(- 10 + 1 i) (- 10 - i)}$

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

$\frac{- 40 - 4 i - 30 i - 3 (i^{1} i)}{(- 10 + 1 i) (- 10 - i)}$

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

$\frac{- 40 - 4 i - 30 i - 3 (i^{1} i^{1})}{(- 10 + 1 i) (- 10 - i)}$

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

$\frac{- 40 - 4 i - 30 i - 3 i^{1 + 1}}{(- 10 + 1 i) (- 10 - i)}$

Add $1$ and $1$ .

$\frac{- 40 - 4 i - 30 i - 3 i^{2}}{(- 10 + 1 i) (- 10 - i)}$

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

$\frac{- 40 - 4 i - 30 i - 3 \cdot - 1}{(- 10 + 1 i) (- 10 - i)}$

Multiply $- 3$ by $- 1$ .

$\frac{- 40 - 4 i - 30 i + 3}{(- 10 + 1 i) (- 10 - i)}$

Add $- 40$ and $3$ .

$\frac{- 37 - 4 i - 30 i}{(- 10 + 1 i) (- 10 - i)}$

Subtract $30 i$ from $- 4 i$ .

$\frac{- 37 - 34 i}{(- 10 + 1 i) (- 10 - i)}$

Simplify the denominator.

Expand $(- 10 + 1 i) (- 10 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 37 - 34 i}{- 10 (- 10 - i) + 1 i (- 10 - i)}$

Apply the distributive property.

$\frac{- 37 - 34 i}{- 10 \cdot - 10 - 10 (- i) + 1 i (- 10 - i)}$

Apply the distributive property.

$\frac{- 37 - 34 i}{- 10 \cdot - 10 - 10 (- i) + 1 i \cdot - 10 + 1 i (- i)}$

Simplify.

Multiply $- 10$ by $- 10$ .

$\frac{- 37 - 34 i}{100 - 10 (- i) + 1 i \cdot - 10 + 1 i (- i)}$

Multiply $- 1$ by $- 10$ .

$\frac{- 37 - 34 i}{100 + 10 i + 1 i \cdot - 10 + 1 i (- i)}$

Multiply $- 10$ by $1$ .

$\frac{- 37 - 34 i}{100 + 10 i - 10 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{- 37 - 34 i}{100 + 10 i - 10 i - i i}$

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

$\frac{- 37 - 34 i}{100 + 10 i - 10 i - (i^{1} i)}$

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

$\frac{- 37 - 34 i}{100 + 10 i - 10 i - (i^{1} i^{1})}$

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

$\frac{- 37 - 34 i}{100 + 10 i - 10 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 37 - 34 i}{100 + 10 i - 10 i - i^{2}}$

Subtract $10 i$ from $10 i$ .

$\frac{- 37 - 34 i}{100 + 0 - i^{2}}$

Add $100$ and $0$ .

$\frac{- 37 - 34 i}{100 - i^{2}}$

Simplify each term.

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

$\frac{- 37 - 34 i}{100 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{- 37 - 34 i}{100 + 1}$

Add $100$ and $1$ .

$\frac{- 37 - 34 i}{101}$

Step 3

Split the fraction $\frac{- 37 - 34 i}{101}$ into two fractions.

$\frac{- 37}{101} + \frac{- 34 i}{101}$

Step 4

Simplify each term.

Move the negative in front of the fraction.

$- \frac{37}{101} + \frac{- 34 i}{101}$

Move the negative in front of the fraction.

$- \frac{37}{101} - \frac{34 i}{101}$

Divide (4+3i)/(-10+i)

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