Simplify (4-2i)/(7+3i)

$\frac{4 - 2 i}{7 + 3 i}$

Multiply the numerator and denominator of $\frac{4 - 2 i}{7 + 3 i}$ by the conjugate of $7 + 3 i$ to make the denominator real.

$\frac{4 - 2 i}{7 + 3 i} \cdot \frac{7 - 3 i}{7 - 3 i}$

Multiply.

Combine.

$\frac{(4 - 2 i) (7 - 3 i)}{(7 + 3 i) (7 - 3 i)}$

Simplify the numerator.

Expand $(4 - 2 i) (7 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{4 (7 - 3 i) - 2 i (7 - 3 i)}{(7 + 3 i) (7 - 3 i)}$

Apply the distributive property.

$\frac{4 \cdot 7 + 4 (- 3 i) - 2 i (7 - 3 i)}{(7 + 3 i) (7 - 3 i)}$

Apply the distributive property.

$\frac{4 \cdot 7 + 4 (- 3 i) - 2 i \cdot 7 - 2 i (- 3 i)}{(7 + 3 i) (7 - 3 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $7$ .

$\frac{28 + 4 (- 3 i) - 2 i \cdot 7 - 2 i (- 3 i)}{(7 + 3 i) (7 - 3 i)}$

Multiply $- 3$ by $4$ .

$\frac{28 - 12 i - 2 i \cdot 7 - 2 i (- 3 i)}{(7 + 3 i) (7 - 3 i)}$

Multiply $7$ by $- 2$ .

$\frac{28 - 12 i - 14 i - 2 i (- 3 i)}{(7 + 3 i) (7 - 3 i)}$

Multiply $- 2 i (- 3 i)$ .

Multiply $- 3$ by $- 2$ .

$\frac{28 - 12 i - 14 i + 6 i i}{(7 + 3 i) (7 - 3 i)}$

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

$\frac{28 - 12 i - 14 i + 6 (i^{1} i)}{(7 + 3 i) (7 - 3 i)}$

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

$\frac{28 - 12 i - 14 i + 6 (i^{1} i^{1})}{(7 + 3 i) (7 - 3 i)}$

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

$\frac{28 - 12 i - 14 i + 6 i^{1 + 1}}{(7 + 3 i) (7 - 3 i)}$

Add $1$ and $1$ .

$\frac{28 - 12 i - 14 i + 6 i^{2}}{(7 + 3 i) (7 - 3 i)}$

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

$\frac{28 - 12 i - 14 i + 6 \cdot - 1}{(7 + 3 i) (7 - 3 i)}$

Multiply $6$ by $- 1$ .

$\frac{28 - 12 i - 14 i - 6}{(7 + 3 i) (7 - 3 i)}$

Subtract $6$ from $28$ .

$\frac{22 - 12 i - 14 i}{(7 + 3 i) (7 - 3 i)}$

Subtract $14 i$ from $- 12 i$ .

$\frac{22 - 26 i}{(7 + 3 i) (7 - 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{22 - 26 i}{7 (7 - 3 i) + 3 i (7 - 3 i)}$

Apply the distributive property.

$\frac{22 - 26 i}{7 \cdot 7 + 7 (- 3 i) + 3 i (7 - 3 i)}$

Apply the distributive property.

$\frac{22 - 26 i}{7 \cdot 7 + 7 (- 3 i) + 3 i \cdot 7 + 3 i (- 3 i)}$

Simplify.

Multiply $7$ by $7$ .

$\frac{22 - 26 i}{49 + 7 (- 3 i) + 3 i \cdot 7 + 3 i (- 3 i)}$

Multiply $- 3$ by $7$ .

$\frac{22 - 26 i}{49 - 21 i + 3 i \cdot 7 + 3 i (- 3 i)}$

Multiply $7$ by $3$ .

$\frac{22 - 26 i}{49 - 21 i + 21 i + 3 i (- 3 i)}$

Multiply $- 3$ by $3$ .

$\frac{22 - 26 i}{49 - 21 i + 21 i - 9 i i}$

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

$\frac{22 - 26 i}{49 - 21 i + 21 i - 9 (i^{1} i)}$

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

$\frac{22 - 26 i}{49 - 21 i + 21 i - 9 (i^{1} i^{1})}$

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

$\frac{22 - 26 i}{49 - 21 i + 21 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{22 - 26 i}{49 - 21 i + 21 i - 9 i^{2}}$

Add $- 21 i$ and $21 i$ .

$\frac{22 - 26 i}{49 + 0 - 9 i^{2}}$

Add $49$ and $0$ .

$\frac{22 - 26 i}{49 - 9 i^{2}}$

Simplify each term.

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

$\frac{22 - 26 i}{49 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{22 - 26 i}{49 + 9}$

Add $49$ and $9$ .

$\frac{22 - 26 i}{58}$

Cancel the common factor of $22 - 26 i$ and $58$ .

Factor $2$ out of $22$ .

$\frac{2 (11) - 26 i}{58}$

Factor $2$ out of $- 26 i$ .

$\frac{2 (11) + 2 (- 13 i)}{58}$

Factor $2$ out of $2 (11) + 2 (- 13 i)$ .

$\frac{2 (11 - 13 i)}{58}$

Cancel the common factors.

Factor $2$ out of $58$ .

$\frac{2 (11 - 13 i)}{2 \cdot 29}$

Cancel the common factor.

$\frac{2 (11 - 13 i)}{2 \cdot 29}$

Rewrite the expression.

$\frac{11 - 13 i}{29}$

Split the fraction $\frac{11 - 13 i}{29}$ into two fractions.

$\frac{11}{29} + \frac{- 13 i}{29}$

Move the negative in front of the fraction.

$\frac{11}{29} - \frac{13 i}{29}$

Simplify (4-2i)/(7+3i)

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