Write in Standard Form (9-9i)/(5+3i)

$\frac{9 - 9 i}{5 + 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{9 - 9 i}{5 + 3 i}$ by the conjugate of $5 + 3 i$ to make the denominator real.

$\frac{9 - 9 i}{5 + 3 i} \cdot \frac{5 - 3 i}{5 - 3 i}$

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(9 - 9 i) (5 - 3 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $5$ .

$\frac{45 + 9 (- 3 i) - 9 i \cdot 5 - 9 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $- 3$ by $9$ .

$\frac{45 - 27 i - 9 i \cdot 5 - 9 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $5$ by $- 9$ .

$\frac{45 - 27 i - 45 i - 9 i (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $- 9 i (- 3 i)$ .

Multiply $- 3$ by $- 9$ .

$\frac{45 - 27 i - 45 i + 27 i i}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{45 - 27 i - 45 i + 27 (i^{1} i)}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{45 - 27 i - 45 i + 27 (i^{1} i^{1})}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{45 - 27 i - 45 i + 27 i^{1 + 1}}{(5 + 3 i) (5 - 3 i)}$

Add $1$ and $1$ .

$\frac{45 - 27 i - 45 i + 27 i^{2}}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{45 - 27 i - 45 i + 27 \cdot - 1}{(5 + 3 i) (5 - 3 i)}$

Multiply $27$ by $- 1$ .

$\frac{45 - 27 i - 45 i - 27}{(5 + 3 i) (5 - 3 i)}$

Subtract $27$ from $45$ .

$\frac{18 - 27 i - 45 i}{(5 + 3 i) (5 - 3 i)}$

Subtract $45 i$ from $- 27 i$ .

$\frac{18 - 72 i}{(5 + 3 i) (5 - 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{18 - 72 i}{5 (5 - 3 i) + 3 i (5 - 3 i)}$

Apply the distributive property.

$\frac{18 - 72 i}{5 \cdot 5 + 5 (- 3 i) + 3 i (5 - 3 i)}$

Apply the distributive property.

$\frac{18 - 72 i}{5 \cdot 5 + 5 (- 3 i) + 3 i \cdot 5 + 3 i (- 3 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{18 - 72 i}{25 + 5 (- 3 i) + 3 i \cdot 5 + 3 i (- 3 i)}$

Multiply $- 3$ by $5$ .

$\frac{18 - 72 i}{25 - 15 i + 3 i \cdot 5 + 3 i (- 3 i)}$

Multiply $5$ by $3$ .

$\frac{18 - 72 i}{25 - 15 i + 15 i + 3 i (- 3 i)}$

Multiply $- 3$ by $3$ .

$\frac{18 - 72 i}{25 - 15 i + 15 i - 9 i i}$

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

$\frac{18 - 72 i}{25 - 15 i + 15 i - 9 (i^{1} i)}$

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

$\frac{18 - 72 i}{25 - 15 i + 15 i - 9 (i^{1} i^{1})}$

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

$\frac{18 - 72 i}{25 - 15 i + 15 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{18 - 72 i}{25 - 15 i + 15 i - 9 i^{2}}$

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

$\frac{18 - 72 i}{25 + 0 - 9 i^{2}}$

Add $25$ and $0$ .

$\frac{18 - 72 i}{25 - 9 i^{2}}$

Simplify each term.

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

$\frac{18 - 72 i}{25 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{18 - 72 i}{25 + 9}$

Add $25$ and $9$ .

$\frac{18 - 72 i}{34}$

Step 3

Cancel the common factor of $18 - 72 i$ and $34$ .

Factor $2$ out of $18$ .

$\frac{2 (9) - 72 i}{34}$

Factor $2$ out of $- 72 i$ .

$\frac{2 (9) + 2 (- 36 i)}{34}$

Factor $2$ out of $2 (9) + 2 (- 36 i)$ .

$\frac{2 (9 - 36 i)}{34}$

Cancel the common factors.

Factor $2$ out of $34$ .

$\frac{2 (9 - 36 i)}{2 \cdot 17}$

Cancel the common factor.

$\frac{2 (9 - 36 i)}{2 \cdot 17}$

Rewrite the expression.

$\frac{9 - 36 i}{17}$

Step 4

Split the fraction $\frac{9 - 36 i}{17}$ into two fractions.

$\frac{9}{17} + \frac{- 36 i}{17}$

Step 5

Move the negative in front of the fraction.

$\frac{9}{17} - \frac{36 i}{17}$

Write in Standard Form (9-9i)/(5+3i)

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