Divide 15/(3+4i)

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

Step 1

Multiply the numerator and denominator of $\frac{15}{3 + 4 i}$ by the conjugate of $3 + 4 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $15$ by $3$ .

$\frac{45 + 15 (- 4 i)}{(3 + 4 i) (3 - 4 i)}$

Multiply $- 4$ by $15$ .

$\frac{45 - 60 i}{(3 + 4 i) (3 - 4 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{45 - 60 i}{3 (3 - 4 i) + 4 i (3 - 4 i)}$

Apply the distributive property.

$\frac{45 - 60 i}{3 \cdot 3 + 3 (- 4 i) + 4 i (3 - 4 i)}$

Apply the distributive property.

$\frac{45 - 60 i}{3 \cdot 3 + 3 (- 4 i) + 4 i \cdot 3 + 4 i (- 4 i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{45 - 60 i}{9 + 3 (- 4 i) + 4 i \cdot 3 + 4 i (- 4 i)}$

Multiply $- 4$ by $3$ .

$\frac{45 - 60 i}{9 - 12 i + 4 i \cdot 3 + 4 i (- 4 i)}$

Multiply $3$ by $4$ .

$\frac{45 - 60 i}{9 - 12 i + 12 i + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

$\frac{45 - 60 i}{9 - 12 i + 12 i - 16 i i}$

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

$\frac{45 - 60 i}{9 - 12 i + 12 i - 16 (i^{1} i)}$

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

$\frac{45 - 60 i}{9 - 12 i + 12 i - 16 (i^{1} i^{1})}$

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

$\frac{45 - 60 i}{9 - 12 i + 12 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{45 - 60 i}{9 - 12 i + 12 i - 16 i^{2}}$

Add $- 12 i$ and $12 i$ .

$\frac{45 - 60 i}{9 + 0 - 16 i^{2}}$

Add $9$ and $0$ .

$\frac{45 - 60 i}{9 - 16 i^{2}}$

Simplify each term.

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

$\frac{45 - 60 i}{9 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{45 - 60 i}{9 + 16}$

Add $9$ and $16$ .

$\frac{45 - 60 i}{25}$

Step 3

Cancel the common factor of $45 - 60 i$ and $25$ .

Factor $5$ out of $45$ .

$\frac{5 (9) - 60 i}{25}$

Factor $5$ out of $- 60 i$ .

$\frac{5 (9) + 5 (- 12 i)}{25}$

Factor $5$ out of $5 (9) + 5 (- 12 i)$ .

$\frac{5 (9 - 12 i)}{25}$

Cancel the common factors.

Factor $5$ out of $25$ .

$\frac{5 (9 - 12 i)}{5 \cdot 5}$

Cancel the common factor.

$\frac{5 (9 - 12 i)}{5 \cdot 5}$

Rewrite the expression.

$\frac{9 - 12 i}{5}$

Step 4

Split the fraction $\frac{9 - 12 i}{5}$ into two fractions.

$\frac{9}{5} + \frac{- 12 i}{5}$

Step 5

Move the negative in front of the fraction.

$\frac{9}{5} - \frac{12 i}{5}$

Divide 15/(3+4i)

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