Write in Standard Form (3+7i)/(5-3i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $5$ .

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

Multiply $3$ by $3$ .

$\frac{15 + 9 i + 7 i \cdot 5 + 7 i (3 i)}{(5 - 3 i) (5 + 3 i)}$

Multiply $5$ by $7$ .

$\frac{15 + 9 i + 35 i + 7 i (3 i)}{(5 - 3 i) (5 + 3 i)}$

Multiply $7 i (3 i)$ .

Multiply $3$ by $7$ .

$\frac{15 + 9 i + 35 i + 21 i i}{(5 - 3 i) (5 + 3 i)}$

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

$\frac{15 + 9 i + 35 i + 21 (i^{1} i)}{(5 - 3 i) (5 + 3 i)}$

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

$\frac{15 + 9 i + 35 i + 21 (i^{1} i^{1})}{(5 - 3 i) (5 + 3 i)}$

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

$\frac{15 + 9 i + 35 i + 21 i^{1 + 1}}{(5 - 3 i) (5 + 3 i)}$

Add $1$ and $1$ .

$\frac{15 + 9 i + 35 i + 21 i^{2}}{(5 - 3 i) (5 + 3 i)}$

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

$\frac{15 + 9 i + 35 i + 21 \cdot - 1}{(5 - 3 i) (5 + 3 i)}$

Multiply $21$ by $- 1$ .

$\frac{15 + 9 i + 35 i - 21}{(5 - 3 i) (5 + 3 i)}$

Subtract $21$ from $15$ .

$\frac{- 6 + 9 i + 35 i}{(5 - 3 i) (5 + 3 i)}$

Add $9 i$ and $35 i$ .

$\frac{- 6 + 44 i}{(5 - 3 i) (5 + 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 6 + 44 i}{5 (5 + 3 i) - 3 i (5 + 3 i)}$

Apply the distributive property.

$\frac{- 6 + 44 i}{5 \cdot 5 + 5 (3 i) - 3 i (5 + 3 i)}$

Apply the distributive property.

$\frac{- 6 + 44 i}{5 \cdot 5 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{- 6 + 44 i}{25 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)}$

Multiply $3$ by $5$ .

$\frac{- 6 + 44 i}{25 + 15 i - 3 i \cdot 5 - 3 i (3 i)}$

Multiply $5$ by $- 3$ .

$\frac{- 6 + 44 i}{25 + 15 i - 15 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{- 6 + 44 i}{25 + 15 i - 15 i - 9 i i}$

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

$\frac{- 6 + 44 i}{25 + 15 i - 15 i - 9 (i^{1} i)}$

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

$\frac{- 6 + 44 i}{25 + 15 i - 15 i - 9 (i^{1} i^{1})}$

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

$\frac{- 6 + 44 i}{25 + 15 i - 15 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 6 + 44 i}{25 + 15 i - 15 i - 9 i^{2}}$

Subtract $15 i$ from $15 i$ .

$\frac{- 6 + 44 i}{25 + 0 - 9 i^{2}}$

Add $25$ and $0$ .

$\frac{- 6 + 44 i}{25 - 9 i^{2}}$

Simplify each term.

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

$\frac{- 6 + 44 i}{25 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{- 6 + 44 i}{25 + 9}$

Add $25$ and $9$ .

$\frac{- 6 + 44 i}{34}$

Step 3

Cancel the common factor of $- 6 + 44 i$ and $34$ .

Factor $2$ out of $- 6$ .

$\frac{2 (- 3) + 44 i}{34}$

Factor $2$ out of $44 i$ .

$\frac{2 (- 3) + 2 (22 i)}{34}$

Factor $2$ out of $2 (- 3) + 2 (22 i)$ .

$\frac{2 (- 3 + 22 i)}{34}$

Cancel the common factors.

Factor $2$ out of $34$ .

$\frac{2 (- 3 + 22 i)}{2 \cdot 17}$

Cancel the common factor.

$\frac{2 (- 3 + 22 i)}{2 \cdot 17}$

Rewrite the expression.

$\frac{- 3 + 22 i}{17}$

Step 4

Split the fraction $\frac{- 3 + 22 i}{17}$ into two fractions.

$\frac{- 3}{17} + \frac{22 i}{17}$

Step 5

Move the negative in front of the fraction.

$- \frac{3}{17} + \frac{22 i}{17}$

Write in Standard Form (3+7i)/(5-3i)

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