Divide (3-55i)/(5+7i)

$\frac{3 - 55 i}{5 + 7 i}$

Multiply the numerator and denominator of $\frac{3 - 55 i}{5 + 7 i}$ by the conjugate of $5 + 7 i$ to make the denominator real.

$\frac{3 - 55 i}{5 + 7 i} \cdot \frac{5 - 7 i}{5 - 7 i}$

Multiply.

Combine.

$\frac{(3 - 55 i) (5 - 7 i)}{(5 + 7 i) (5 - 7 i)}$

Simplify the numerator.

Expand $(3 - 55 i) (5 - 7 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{3 (5 - 7 i) - 55 i (5 - 7 i)}{(5 + 7 i) (5 - 7 i)}$

Apply the distributive property.

$\frac{3 \cdot 5 + 3 (- 7 i) - 55 i (5 - 7 i)}{(5 + 7 i) (5 - 7 i)}$

Apply the distributive property.

$\frac{3 \cdot 5 + 3 (- 7 i) - 55 i \cdot 5 - 55 i (- 7 i)}{(5 + 7 i) (5 - 7 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $5$ .

$\frac{15 + 3 (- 7 i) - 55 i \cdot 5 - 55 i (- 7 i)}{(5 + 7 i) (5 - 7 i)}$

Multiply $- 7$ by $3$ .

$\frac{15 - 21 i - 55 i \cdot 5 - 55 i (- 7 i)}{(5 + 7 i) (5 - 7 i)}$

Multiply $5$ by $- 55$ .

$\frac{15 - 21 i - 275 i - 55 i (- 7 i)}{(5 + 7 i) (5 - 7 i)}$

Multiply $- 55 i (- 7 i)$ .

Multiply $- 7$ by $- 55$ .

$\frac{15 - 21 i - 275 i + 385 i i}{(5 + 7 i) (5 - 7 i)}$

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

$\frac{15 - 21 i - 275 i + 385 (i^{1} i)}{(5 + 7 i) (5 - 7 i)}$

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

$\frac{15 - 21 i - 275 i + 385 (i^{1} i^{1})}{(5 + 7 i) (5 - 7 i)}$

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

$\frac{15 - 21 i - 275 i + 385 i^{1 + 1}}{(5 + 7 i) (5 - 7 i)}$

Add $1$ and $1$ .

$\frac{15 - 21 i - 275 i + 385 i^{2}}{(5 + 7 i) (5 - 7 i)}$

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

$\frac{15 - 21 i - 275 i + 385 \cdot - 1}{(5 + 7 i) (5 - 7 i)}$

Multiply $385$ by $- 1$ .

$\frac{15 - 21 i - 275 i - 385}{(5 + 7 i) (5 - 7 i)}$

Subtract $385$ from $15$ .

$\frac{- 370 - 21 i - 275 i}{(5 + 7 i) (5 - 7 i)}$

Subtract $275 i$ from $- 21 i$ .

$\frac{- 370 - 296 i}{(5 + 7 i) (5 - 7 i)}$

Simplify the denominator.

Expand $(5 + 7 i) (5 - 7 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 370 - 296 i}{5 (5 - 7 i) + 7 i (5 - 7 i)}$

Apply the distributive property.

$\frac{- 370 - 296 i}{5 \cdot 5 + 5 (- 7 i) + 7 i (5 - 7 i)}$

Apply the distributive property.

$\frac{- 370 - 296 i}{5 \cdot 5 + 5 (- 7 i) + 7 i \cdot 5 + 7 i (- 7 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{- 370 - 296 i}{25 + 5 (- 7 i) + 7 i \cdot 5 + 7 i (- 7 i)}$

Multiply $- 7$ by $5$ .

$\frac{- 370 - 296 i}{25 - 35 i + 7 i \cdot 5 + 7 i (- 7 i)}$

Multiply $5$ by $7$ .

$\frac{- 370 - 296 i}{25 - 35 i + 35 i + 7 i (- 7 i)}$

Multiply $- 7$ by $7$ .

$\frac{- 370 - 296 i}{25 - 35 i + 35 i - 49 i i}$

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

$\frac{- 370 - 296 i}{25 - 35 i + 35 i - 49 (i^{1} i)}$

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

$\frac{- 370 - 296 i}{25 - 35 i + 35 i - 49 (i^{1} i^{1})}$

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

$\frac{- 370 - 296 i}{25 - 35 i + 35 i - 49 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 370 - 296 i}{25 - 35 i + 35 i - 49 i^{2}}$

Add $- 35 i$ and $35 i$ .

$\frac{- 370 - 296 i}{25 + 0 - 49 i^{2}}$

Add $25$ and $0$ .

$\frac{- 370 - 296 i}{25 - 49 i^{2}}$

Simplify each term.

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

$\frac{- 370 - 296 i}{25 - 49 \cdot - 1}$

Multiply $- 49$ by $- 1$ .

$\frac{- 370 - 296 i}{25 + 49}$

Add $25$ and $49$ .

$\frac{- 370 - 296 i}{74}$

Cancel the common factor of $- 370 - 296 i$ and $74$ .

Factor $74$ out of $- 370$ .

$\frac{74 \cdot - 5 - 296 i}{74}$

Factor $74$ out of $- 296 i$ .

$\frac{74 \cdot - 5 + 74 (- 4 i)}{74}$

Factor $74$ out of $74 (- 5) + 74 (- 4 i)$ .

$\frac{74 (- 5 - 4 i)}{74}$

Cancel the common factors.

Factor $74$ out of $74$ .

$\frac{74 (- 5 - 4 i)}{74 (1)}$

Cancel the common factor.

$\frac{74 (- 5 - 4 i)}{74 \cdot 1}$

Rewrite the expression.

$\frac{- 5 - 4 i}{1}$

Divide $- 5 - 4 i$ by $1$ .

$- 5 - 4 i$

Divide (3-55i)/(5+7i)

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