Write in Standard Form (-2+2i)/(5+3i)

$\frac{- 2 + 2 i}{5 + 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{- 2 + 2 i}{5 + 3 i}$ by the conjugate of $5 + 3 i$ to make the denominator real.

$\frac{- 2 + 2 i}{5 + 3 i} \cdot \frac{5 - 3 i}{5 - 3 i}$

Step 2

Multiply.

Combine.

$\frac{(- 2 + 2 i) (5 - 3 i)}{(5 + 3 i) (5 - 3 i)}$

Simplify the numerator.

Expand $(- 2 + 2 i) (5 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 2 (5 - 3 i) + 2 i (5 - 3 i)}{(5 + 3 i) (5 - 3 i)}$

Apply the distributive property.

$\frac{- 2 \cdot 5 - 2 (- 3 i) + 2 i (5 - 3 i)}{(5 + 3 i) (5 - 3 i)}$

Apply the distributive property.

$\frac{- 2 \cdot 5 - 2 (- 3 i) + 2 i \cdot 5 + 2 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $5$ .

$\frac{- 10 - 2 (- 3 i) + 2 i \cdot 5 + 2 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $- 3$ by $- 2$ .

$\frac{- 10 + 6 i + 2 i \cdot 5 + 2 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $5$ by $2$ .

$\frac{- 10 + 6 i + 10 i + 2 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $2 i (- 3 i)$ .

Multiply $- 3$ by $2$ .

$\frac{- 10 + 6 i + 10 i - 6 i i}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{- 10 + 6 i + 10 i - 6 (i^{1} i)}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{- 10 + 6 i + 10 i - 6 (i^{1} i^{1})}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{- 10 + 6 i + 10 i - 6 i^{1 + 1}}{(5 + 3 i) (5 - 3 i)}$

Add $1$ and $1$ .

$\frac{- 10 + 6 i + 10 i - 6 i^{2}}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{- 10 + 6 i + 10 i - 6 \cdot - 1}{(5 + 3 i) (5 - 3 i)}$

Multiply $- 6$ by $- 1$ .

$\frac{- 10 + 6 i + 10 i + 6}{(5 + 3 i) (5 - 3 i)}$

Add $- 10$ and $6$ .

$\frac{- 4 + 6 i + 10 i}{(5 + 3 i) (5 - 3 i)}$

Add $6 i$ and $10 i$ .

$\frac{- 4 + 16 i}{(5 + 3 i) (5 - 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 4 + 16 i}{5 (5 - 3 i) + 3 i (5 - 3 i)}$

Apply the distributive property.

$\frac{- 4 + 16 i}{5 \cdot 5 + 5 (- 3 i) + 3 i (5 - 3 i)}$

Apply the distributive property.

$\frac{- 4 + 16 i}{5 \cdot 5 + 5 (- 3 i) + 3 i \cdot 5 + 3 i (- 3 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{- 4 + 16 i}{25 + 5 (- 3 i) + 3 i \cdot 5 + 3 i (- 3 i)}$

Multiply $- 3$ by $5$ .

$\frac{- 4 + 16 i}{25 - 15 i + 3 i \cdot 5 + 3 i (- 3 i)}$

Multiply $5$ by $3$ .

$\frac{- 4 + 16 i}{25 - 15 i + 15 i + 3 i (- 3 i)}$

Multiply $- 3$ by $3$ .

$\frac{- 4 + 16 i}{25 - 15 i + 15 i - 9 i i}$

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

$\frac{- 4 + 16 i}{25 - 15 i + 15 i - 9 (i^{1} i)}$

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

$\frac{- 4 + 16 i}{25 - 15 i + 15 i - 9 (i^{1} i^{1})}$

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

$\frac{- 4 + 16 i}{25 - 15 i + 15 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 4 + 16 i}{25 - 15 i + 15 i - 9 i^{2}}$

Add $- 15 i$ and $15 i$ .

$\frac{- 4 + 16 i}{25 + 0 - 9 i^{2}}$

Add $25$ and $0$ .

$\frac{- 4 + 16 i}{25 - 9 i^{2}}$

Simplify each term.

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

$\frac{- 4 + 16 i}{25 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{- 4 + 16 i}{25 + 9}$

Add $25$ and $9$ .

$\frac{- 4 + 16 i}{34}$

Step 3

Cancel the common factor of $- 4 + 16 i$ and $34$ .

Factor $2$ out of $- 4$ .

$\frac{2 (- 2) + 16 i}{34}$

Factor $2$ out of $16 i$ .

$\frac{2 (- 2) + 2 (8 i)}{34}$

Factor $2$ out of $2 (- 2) + 2 (8 i)$ .

$\frac{2 (- 2 + 8 i)}{34}$

Cancel the common factors.

Factor $2$ out of $34$ .

$\frac{2 (- 2 + 8 i)}{2 \cdot 17}$

Cancel the common factor.

$\frac{2 (- 2 + 8 i)}{2 \cdot 17}$

Rewrite the expression.

$\frac{- 2 + 8 i}{17}$

Step 4

Split the fraction $\frac{- 2 + 8 i}{17}$ into two fractions.

$\frac{- 2}{17} + \frac{8 i}{17}$

Step 5

Move the negative in front of the fraction.

$- \frac{2}{17} + \frac{8 i}{17}$

Write in Standard Form (-2+2i)/(5+3i)

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