Write in Standard Form (9-4i)/(5+7i)

$\frac{9 - 4 i}{5 + 7 i}$

Step 1

Multiply the numerator and denominator of $\frac{9 - 4 i}{5 + 7 i}$ by the conjugate of $5 + 7 i$ to make the denominator real.

$\frac{9 - 4 i}{5 + 7 i} \cdot \frac{5 - 7 i}{5 - 7 i}$

Step 2

Multiply.

Combine.

$\frac{(9 - 4 i) (5 - 7 i)}{(5 + 7 i) (5 - 7 i)}$

Simplify the numerator.

Expand $(9 - 4 i) (5 - 7 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{9 (5 - 7 i) - 4 i (5 - 7 i)}{(5 + 7 i) (5 - 7 i)}$

Apply the distributive property.

$\frac{9 \cdot 5 + 9 (- 7 i) - 4 i (5 - 7 i)}{(5 + 7 i) (5 - 7 i)}$

Apply the distributive property.

$\frac{9 \cdot 5 + 9 (- 7 i) - 4 i \cdot 5 - 4 i (- 7 i)}{(5 + 7 i) (5 - 7 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $5$ .

$\frac{45 + 9 (- 7 i) - 4 i \cdot 5 - 4 i (- 7 i)}{(5 + 7 i) (5 - 7 i)}$

Multiply $- 7$ by $9$ .

$\frac{45 - 63 i - 4 i \cdot 5 - 4 i (- 7 i)}{(5 + 7 i) (5 - 7 i)}$

Multiply $5$ by $- 4$ .

$\frac{45 - 63 i - 20 i - 4 i (- 7 i)}{(5 + 7 i) (5 - 7 i)}$

Multiply $- 4 i (- 7 i)$ .

Multiply $- 7$ by $- 4$ .

$\frac{45 - 63 i - 20 i + 28 i i}{(5 + 7 i) (5 - 7 i)}$

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

$\frac{45 - 63 i - 20 i + 28 (i^{1} i)}{(5 + 7 i) (5 - 7 i)}$

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

$\frac{45 - 63 i - 20 i + 28 (i^{1} i^{1})}{(5 + 7 i) (5 - 7 i)}$

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

$\frac{45 - 63 i - 20 i + 28 i^{1 + 1}}{(5 + 7 i) (5 - 7 i)}$

Add $1$ and $1$ .

$\frac{45 - 63 i - 20 i + 28 i^{2}}{(5 + 7 i) (5 - 7 i)}$

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

$\frac{45 - 63 i - 20 i + 28 \cdot - 1}{(5 + 7 i) (5 - 7 i)}$

Multiply $28$ by $- 1$ .

$\frac{45 - 63 i - 20 i - 28}{(5 + 7 i) (5 - 7 i)}$

Subtract $28$ from $45$ .

$\frac{17 - 63 i - 20 i}{(5 + 7 i) (5 - 7 i)}$

Subtract $20 i$ from $- 63 i$ .

$\frac{17 - 83 i}{(5 + 7 i) (5 - 7 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{17 - 83 i}{5 (5 - 7 i) + 7 i (5 - 7 i)}$

Apply the distributive property.

$\frac{17 - 83 i}{5 \cdot 5 + 5 (- 7 i) + 7 i (5 - 7 i)}$

Apply the distributive property.

$\frac{17 - 83 i}{5 \cdot 5 + 5 (- 7 i) + 7 i \cdot 5 + 7 i (- 7 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{17 - 83 i}{25 + 5 (- 7 i) + 7 i \cdot 5 + 7 i (- 7 i)}$

Multiply $- 7$ by $5$ .

$\frac{17 - 83 i}{25 - 35 i + 7 i \cdot 5 + 7 i (- 7 i)}$

Multiply $5$ by $7$ .

$\frac{17 - 83 i}{25 - 35 i + 35 i + 7 i (- 7 i)}$

Multiply $- 7$ by $7$ .

$\frac{17 - 83 i}{25 - 35 i + 35 i - 49 i i}$

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

$\frac{17 - 83 i}{25 - 35 i + 35 i - 49 (i^{1} i)}$

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

$\frac{17 - 83 i}{25 - 35 i + 35 i - 49 (i^{1} i^{1})}$

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

$\frac{17 - 83 i}{25 - 35 i + 35 i - 49 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{17 - 83 i}{25 - 35 i + 35 i - 49 i^{2}}$

Add $- 35 i$ and $35 i$ .

$\frac{17 - 83 i}{25 + 0 - 49 i^{2}}$

Add $25$ and $0$ .

$\frac{17 - 83 i}{25 - 49 i^{2}}$

Simplify each term.

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

$\frac{17 - 83 i}{25 - 49 \cdot - 1}$

Multiply $- 49$ by $- 1$ .

$\frac{17 - 83 i}{25 + 49}$

Add $25$ and $49$ .

$\frac{17 - 83 i}{74}$

Step 3

Split the fraction $\frac{17 - 83 i}{74}$ into two fractions.

$\frac{17}{74} + \frac{- 83 i}{74}$

Step 4

Move the negative in front of the fraction.

$\frac{17}{74} - \frac{83 i}{74}$

Write in Standard Form (9-4i)/(5+7i)

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