Divide (2i)/(3-9i)

$\frac{2 i}{3 - 9 i}$

Step 1

Multiply the numerator and denominator of $\frac{2 i}{3 - 9 i}$ by the conjugate of $3 - 9 i$ to make the denominator real.

$\frac{2 i}{3 - 9 i} \cdot \frac{3 + 9 i}{3 + 9 i}$

Step 2

Multiply.

Combine.

$\frac{2 i (3 + 9 i)}{(3 - 9 i) (3 + 9 i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{2 i \cdot 3 + 2 i (9 i)}{(3 - 9 i) (3 + 9 i)}$

Multiply $3$ by $2$ .

$\frac{6 i + 2 i (9 i)}{(3 - 9 i) (3 + 9 i)}$

Multiply $2 i (9 i)$ .

Multiply $9$ by $2$ .

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

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

$\frac{6 i + 18 (i^{1} i)}{(3 - 9 i) (3 + 9 i)}$

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

$\frac{6 i + 18 (i^{1} i^{1})}{(3 - 9 i) (3 + 9 i)}$

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

$\frac{6 i + 18 i^{1 + 1}}{(3 - 9 i) (3 + 9 i)}$

Add $1$ and $1$ .

$\frac{6 i + 18 i^{2}}{(3 - 9 i) (3 + 9 i)}$

Simplify each term.

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

$\frac{6 i + 18 \cdot - 1}{(3 - 9 i) (3 + 9 i)}$

Multiply $18$ by $- 1$ .

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

Reorder $6 i$ and $- 18$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $3$ by $3$ .

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

Multiply $9$ by $3$ .

$\frac{- 18 + 6 i}{9 + 27 i - 9 i \cdot 3 - 9 i (9 i)}$

Multiply $3$ by $- 9$ .

$\frac{- 18 + 6 i}{9 + 27 i - 27 i - 9 i (9 i)}$

Multiply $9$ by $- 9$ .

$\frac{- 18 + 6 i}{9 + 27 i - 27 i - 81 i i}$

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

$\frac{- 18 + 6 i}{9 + 27 i - 27 i - 81 (i^{1} i)}$

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

$\frac{- 18 + 6 i}{9 + 27 i - 27 i - 81 (i^{1} i^{1})}$

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

$\frac{- 18 + 6 i}{9 + 27 i - 27 i - 81 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 18 + 6 i}{9 + 27 i - 27 i - 81 i^{2}}$

Subtract $27 i$ from $27 i$ .

$\frac{- 18 + 6 i}{9 + 0 - 81 i^{2}}$

Add $9$ and $0$ .

$\frac{- 18 + 6 i}{9 - 81 i^{2}}$

Simplify each term.

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

$\frac{- 18 + 6 i}{9 - 81 \cdot - 1}$

Multiply $- 81$ by $- 1$ .

$\frac{- 18 + 6 i}{9 + 81}$

Add $9$ and $81$ .

$\frac{- 18 + 6 i}{90}$

Step 3

Cancel the common factor of $- 18 + 6 i$ and $90$ .

Factor $6$ out of $- 18$ .

$\frac{6 \cdot - 3 + 6 i}{90}$

Factor $6$ out of $6 i$ .

$\frac{6 \cdot - 3 + 6 (i)}{90}$

Factor $6$ out of $6 \cdot - 3 + 6 (i)$ .

$\frac{6 \cdot (- 3 + i)}{90}$

Cancel the common factors.

Factor $6$ out of $90$ .

$\frac{6 \cdot (- 3 + i)}{6 (15)}$

Cancel the common factor.

$\frac{6 \cdot (- 3 + i)}{6 \cdot 15}$

Rewrite the expression.

$\frac{- 3 + i}{15}$

Step 4

Split the fraction $\frac{- 3 + i}{15}$ into two fractions.

$\frac{- 3}{15} + \frac{i}{15}$

Step 5

Simplify each term.

Cancel the common factor of $- 3$ and $15$ .

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{15} + \frac{i}{15}$

Cancel the common factors.

Factor $3$ out of $15$ .

$\frac{3 \cdot - 1}{3 \cdot 5} + \frac{i}{15}$

Cancel the common factor.

$\frac{3 \cdot - 1}{3 \cdot 5} + \frac{i}{15}$

Rewrite the expression.

$\frac{- 1}{5} + \frac{i}{15}$

Move the negative in front of the fraction.

$- \frac{1}{5} + \frac{i}{15}$

Divide (2i)/(3-9i)

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