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

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

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

Multiply $3$ by $7$ .

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

Multiply $7$ by $3$ .

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

Multiply $3 i (3 i)$ .

Multiply $3$ by $3$ .

$\frac{49 + 21 i + 21 i + 9 i i}{(7 - 3 i) (7 + 3 i)}$

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

$\frac{49 + 21 i + 21 i + 9 (i^{1} i)}{(7 - 3 i) (7 + 3 i)}$

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

$\frac{49 + 21 i + 21 i + 9 (i^{1} i^{1})}{(7 - 3 i) (7 + 3 i)}$

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

$\frac{49 + 21 i + 21 i + 9 i^{1 + 1}}{(7 - 3 i) (7 + 3 i)}$

Add $1$ and $1$ .

$\frac{49 + 21 i + 21 i + 9 i^{2}}{(7 - 3 i) (7 + 3 i)}$

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

$\frac{49 + 21 i + 21 i + 9 \cdot - 1}{(7 - 3 i) (7 + 3 i)}$

Multiply $9$ by $- 1$ .

$\frac{49 + 21 i + 21 i - 9}{(7 - 3 i) (7 + 3 i)}$

Subtract $9$ from $49$ .

$\frac{40 + 21 i + 21 i}{(7 - 3 i) (7 + 3 i)}$

Add $21 i$ and $21 i$ .

$\frac{40 + 42 i}{(7 - 3 i) (7 + 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{40 + 42 i}{7 (7 + 3 i) - 3 i (7 + 3 i)}$

Apply the distributive property.

$\frac{40 + 42 i}{7 \cdot 7 + 7 (3 i) - 3 i (7 + 3 i)}$

Apply the distributive property.

$\frac{40 + 42 i}{7 \cdot 7 + 7 (3 i) - 3 i \cdot 7 - 3 i (3 i)}$

Simplify.

Multiply $7$ by $7$ .

$\frac{40 + 42 i}{49 + 7 (3 i) - 3 i \cdot 7 - 3 i (3 i)}$

Multiply $3$ by $7$ .

$\frac{40 + 42 i}{49 + 21 i - 3 i \cdot 7 - 3 i (3 i)}$

Multiply $7$ by $- 3$ .

$\frac{40 + 42 i}{49 + 21 i - 21 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{40 + 42 i}{49 + 21 i - 21 i - 9 i i}$

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

$\frac{40 + 42 i}{49 + 21 i - 21 i - 9 (i^{1} i)}$

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

$\frac{40 + 42 i}{49 + 21 i - 21 i - 9 (i^{1} i^{1})}$

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

$\frac{40 + 42 i}{49 + 21 i - 21 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{40 + 42 i}{49 + 21 i - 21 i - 9 i^{2}}$

Subtract $21 i$ from $21 i$ .

$\frac{40 + 42 i}{49 + 0 - 9 i^{2}}$

Add $49$ and $0$ .

$\frac{40 + 42 i}{49 - 9 i^{2}}$

Simplify each term.

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

$\frac{40 + 42 i}{49 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{40 + 42 i}{49 + 9}$

Add $49$ and $9$ .

$\frac{40 + 42 i}{58}$

Step 3

Cancel the common factor of $40 + 42 i$ and $58$ .

Factor $2$ out of $40$ .

$\frac{2 (20) + 42 i}{58}$

Factor $2$ out of $42 i$ .

$\frac{2 (20) + 2 (21 i)}{58}$

Factor $2$ out of $2 (20) + 2 (21 i)$ .

$\frac{2 (20 + 21 i)}{58}$

Cancel the common factors.

Factor $2$ out of $58$ .

$\frac{2 (20 + 21 i)}{2 \cdot 29}$

Cancel the common factor.

$\frac{2 (20 + 21 i)}{2 \cdot 29}$

Rewrite the expression.

$\frac{20 + 21 i}{29}$

Step 4

Split the fraction $\frac{20 + 21 i}{29}$ into two fractions.

$\frac{20}{29} + \frac{21 i}{29}$

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

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