Write in Standard Form (16i-4)/(3i+5)

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

Step 1

Multiply the numerator and denominator of $\frac{16 i - 4}{5 + 3 i}$ by the conjugate of $5 + 3 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(16 i - 4) (5 - 3 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $16$ .

$\frac{80 i + 16 i (- 3 i) - 4 \cdot 5 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $16 i (- 3 i)$ .

Multiply $- 3$ by $16$ .

$\frac{80 i - 48 i i - 4 \cdot 5 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{80 i - 48 (i^{1} i) - 4 \cdot 5 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{80 i - 48 (i^{1} i^{1}) - 4 \cdot 5 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{80 i - 48 i^{1 + 1} - 4 \cdot 5 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Add $1$ and $1$ .

$\frac{80 i - 48 i^{2} - 4 \cdot 5 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

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

$\frac{80 i - 48 \cdot - 1 - 4 \cdot 5 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $- 48$ by $- 1$ .

$\frac{80 i + 48 - 4 \cdot 5 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $- 4$ by $5$ .

$\frac{80 i + 48 - 20 - 4 (- 3 i)}{(5 + 3 i) (5 - 3 i)}$

Multiply $- 3$ by $- 4$ .

$\frac{80 i + 48 - 20 + 12 i}{(5 + 3 i) (5 - 3 i)}$

Add $80 i$ and $12 i$ .

$\frac{48 - 20 + 92 i}{(5 + 3 i) (5 - 3 i)}$

Subtract $20$ from $48$ .

$\frac{28 + 92 i}{(5 + 3 i) (5 - 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{28 + 92 i}{5 (5 - 3 i) + 3 i (5 - 3 i)}$

Apply the distributive property.

$\frac{28 + 92 i}{5 \cdot 5 + 5 (- 3 i) + 3 i (5 - 3 i)}$

Apply the distributive property.

$\frac{28 + 92 i}{5 \cdot 5 + 5 (- 3 i) + 3 i \cdot 5 + 3 i (- 3 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{28 + 92 i}{25 + 5 (- 3 i) + 3 i \cdot 5 + 3 i (- 3 i)}$

Multiply $- 3$ by $5$ .

$\frac{28 + 92 i}{25 - 15 i + 3 i \cdot 5 + 3 i (- 3 i)}$

Multiply $5$ by $3$ .

$\frac{28 + 92 i}{25 - 15 i + 15 i + 3 i (- 3 i)}$

Multiply $- 3$ by $3$ .

$\frac{28 + 92 i}{25 - 15 i + 15 i - 9 i i}$

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

$\frac{28 + 92 i}{25 - 15 i + 15 i - 9 (i^{1} i)}$

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

$\frac{28 + 92 i}{25 - 15 i + 15 i - 9 (i^{1} i^{1})}$

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

$\frac{28 + 92 i}{25 - 15 i + 15 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{28 + 92 i}{25 - 15 i + 15 i - 9 i^{2}}$

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

$\frac{28 + 92 i}{25 + 0 - 9 i^{2}}$

Add $25$ and $0$ .

$\frac{28 + 92 i}{25 - 9 i^{2}}$

Simplify each term.

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

$\frac{28 + 92 i}{25 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{28 + 92 i}{25 + 9}$

Add $25$ and $9$ .

$\frac{28 + 92 i}{34}$

Step 3

Cancel the common factor of $28 + 92 i$ and $34$ .

Factor $2$ out of $28$ .

$\frac{2 (14) + 92 i}{34}$

Factor $2$ out of $92 i$ .

$\frac{2 (14) + 2 (46 i)}{34}$

Factor $2$ out of $2 (14) + 2 (46 i)$ .

$\frac{2 (14 + 46 i)}{34}$

Cancel the common factors.

Factor $2$ out of $34$ .

$\frac{2 (14 + 46 i)}{2 \cdot 17}$

Cancel the common factor.

$\frac{2 (14 + 46 i)}{2 \cdot 17}$

Rewrite the expression.

$\frac{14 + 46 i}{17}$

Step 4

Split the fraction $\frac{14 + 46 i}{17}$ into two fractions.

$\frac{14}{17} + \frac{46 i}{17}$

Write in Standard Form (16i-4)/(3i+5)

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