Write in Standard Form (4i)/(-3+6i)

$\frac{4 i}{- 3 + 6 i}$

Step 1

Multiply the numerator and denominator of $\frac{4 i}{- 3 + 6 i}$ by the conjugate of $- 3 + 6 i$ to make the denominator real.

$\frac{4 i}{- 3 + 6 i} \cdot \frac{- 3 - 6 i}{- 3 - 6 i}$

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 3$ by $4$ .

$\frac{- 12 i + 4 i (- 6 i)}{(- 3 + 6 i) (- 3 - 6 i)}$

Multiply $4 i (- 6 i)$ .

Multiply $- 6$ by $4$ .

$\frac{- 12 i - 24 i i}{(- 3 + 6 i) (- 3 - 6 i)}$

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

$\frac{- 12 i - 24 (i^{1} i)}{(- 3 + 6 i) (- 3 - 6 i)}$

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

$\frac{- 12 i - 24 (i^{1} i^{1})}{(- 3 + 6 i) (- 3 - 6 i)}$

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

$\frac{- 12 i - 24 i^{1 + 1}}{(- 3 + 6 i) (- 3 - 6 i)}$

Add $1$ and $1$ .

$\frac{- 12 i - 24 i^{2}}{(- 3 + 6 i) (- 3 - 6 i)}$

Simplify each term.

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

$\frac{- 12 i - 24 \cdot - 1}{(- 3 + 6 i) (- 3 - 6 i)}$

Multiply $- 24$ by $- 1$ .

$\frac{- 12 i + 24}{(- 3 + 6 i) (- 3 - 6 i)}$

Reorder $- 12 i$ and $24$ .

$\frac{24 - 12 i}{(- 3 + 6 i) (- 3 - 6 i)}$

Simplify the denominator.

Expand $(- 3 + 6 i) (- 3 - 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{24 - 12 i}{- 3 (- 3 - 6 i) + 6 i (- 3 - 6 i)}$

Apply the distributive property.

$\frac{24 - 12 i}{- 3 \cdot - 3 - 3 (- 6 i) + 6 i (- 3 - 6 i)}$

Apply the distributive property.

$\frac{24 - 12 i}{- 3 \cdot - 3 - 3 (- 6 i) + 6 i \cdot - 3 + 6 i (- 6 i)}$

Simplify.

Multiply $- 3$ by $- 3$ .

$\frac{24 - 12 i}{9 - 3 (- 6 i) + 6 i \cdot - 3 + 6 i (- 6 i)}$

Multiply $- 6$ by $- 3$ .

$\frac{24 - 12 i}{9 + 18 i + 6 i \cdot - 3 + 6 i (- 6 i)}$

Multiply $- 3$ by $6$ .

$\frac{24 - 12 i}{9 + 18 i - 18 i + 6 i (- 6 i)}$

Multiply $- 6$ by $6$ .

$\frac{24 - 12 i}{9 + 18 i - 18 i - 36 i i}$

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

$\frac{24 - 12 i}{9 + 18 i - 18 i - 36 (i^{1} i)}$

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

$\frac{24 - 12 i}{9 + 18 i - 18 i - 36 (i^{1} i^{1})}$

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

$\frac{24 - 12 i}{9 + 18 i - 18 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{24 - 12 i}{9 + 18 i - 18 i - 36 i^{2}}$

Subtract $18 i$ from $18 i$ .

$\frac{24 - 12 i}{9 + 0 - 36 i^{2}}$

Add $9$ and $0$ .

$\frac{24 - 12 i}{9 - 36 i^{2}}$

Simplify each term.

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

$\frac{24 - 12 i}{9 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{24 - 12 i}{9 + 36}$

Add $9$ and $36$ .

$\frac{24 - 12 i}{45}$

Step 3

Cancel the common factor of $24 - 12 i$ and $45$ .

Factor $3$ out of $24$ .

$\frac{3 (8) - 12 i}{45}$

Factor $3$ out of $- 12 i$ .

$\frac{3 (8) + 3 (- 4 i)}{45}$

Factor $3$ out of $3 (8) + 3 (- 4 i)$ .

$\frac{3 (8 - 4 i)}{45}$

Cancel the common factors.

Factor $3$ out of $45$ .

$\frac{3 (8 - 4 i)}{3 \cdot 15}$

Cancel the common factor.

$\frac{3 (8 - 4 i)}{3 \cdot 15}$

Rewrite the expression.

$\frac{8 - 4 i}{15}$

Step 4

Split the fraction $\frac{8 - 4 i}{15}$ into two fractions.

$\frac{8}{15} + \frac{- 4 i}{15}$

Step 5

Move the negative in front of the fraction.

$\frac{8}{15} - \frac{4 i}{15}$

Write in Standard Form (4i)/(-3+6i)

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