Solve for G G=(-2+3i)/(-2-3i)

$G = \frac{- 2 + 3 i}{- 2 - 3 i}$

Step 1

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

$G = \frac{- 2 + 3 i}{- 2 - 3 i} \cdot \frac{- 2 + 3 i}{- 2 + 3 i}$

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(- 2 + 3 i) (- 2 + 3 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $- 2$ .

$G = \frac{4 - 2 (3 i) + 3 i \cdot - 2 + 3 i (3 i)}{(- 2 - 3 i) (- 2 + 3 i)}$

Multiply $3$ by $- 2$ .

$G = \frac{4 - 6 i + 3 i \cdot - 2 + 3 i (3 i)}{(- 2 - 3 i) (- 2 + 3 i)}$

Multiply $- 2$ by $3$ .

$G = \frac{4 - 6 i - 6 i + 3 i (3 i)}{(- 2 - 3 i) (- 2 + 3 i)}$

Multiply $3 i (3 i)$ .

Multiply $3$ by $3$ .

$G = \frac{4 - 6 i - 6 i + 9 i i}{(- 2 - 3 i) (- 2 + 3 i)}$

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

$G = \frac{4 - 6 i - 6 i + 9 (i^{1} i)}{(- 2 - 3 i) (- 2 + 3 i)}$

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

$G = \frac{4 - 6 i - 6 i + 9 (i^{1} i^{1})}{(- 2 - 3 i) (- 2 + 3 i)}$

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

$G = \frac{4 - 6 i - 6 i + 9 i^{1 + 1}}{(- 2 - 3 i) (- 2 + 3 i)}$

Add $1$ and $1$ .

$G = \frac{4 - 6 i - 6 i + 9 i^{2}}{(- 2 - 3 i) (- 2 + 3 i)}$

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

$G = \frac{4 - 6 i - 6 i + 9 \cdot - 1}{(- 2 - 3 i) (- 2 + 3 i)}$

Multiply $9$ by $- 1$ .

$G = \frac{4 - 6 i - 6 i - 9}{(- 2 - 3 i) (- 2 + 3 i)}$

Subtract $9$ from $4$ .

$G = \frac{- 5 - 6 i - 6 i}{(- 2 - 3 i) (- 2 + 3 i)}$

Subtract $6 i$ from $- 6 i$ .

$G = \frac{- 5 - 12 i}{(- 2 - 3 i) (- 2 + 3 i)}$

Simplify the denominator.

Expand $(- 2 - 3 i) (- 2 + 3 i)$ using the FOIL Method.

Apply the distributive property.

$G = \frac{- 5 - 12 i}{- 2 (- 2 + 3 i) - 3 i (- 2 + 3 i)}$

Apply the distributive property.

$G = \frac{- 5 - 12 i}{- 2 \cdot - 2 - 2 (3 i) - 3 i (- 2 + 3 i)}$

Apply the distributive property.

$G = \frac{- 5 - 12 i}{- 2 \cdot - 2 - 2 (3 i) - 3 i \cdot - 2 - 3 i (3 i)}$

Simplify.

Multiply $- 2$ by $- 2$ .

$G = \frac{- 5 - 12 i}{4 - 2 (3 i) - 3 i \cdot - 2 - 3 i (3 i)}$

Multiply $3$ by $- 2$ .

$G = \frac{- 5 - 12 i}{4 - 6 i - 3 i \cdot - 2 - 3 i (3 i)}$

Multiply $- 2$ by $- 3$ .

$G = \frac{- 5 - 12 i}{4 - 6 i + 6 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$G = \frac{- 5 - 12 i}{4 - 6 i + 6 i - 9 i i}$

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

$G = \frac{- 5 - 12 i}{4 - 6 i + 6 i - 9 (i^{1} i)}$

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

$G = \frac{- 5 - 12 i}{4 - 6 i + 6 i - 9 (i^{1} i^{1})}$

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

$G = \frac{- 5 - 12 i}{4 - 6 i + 6 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$G = \frac{- 5 - 12 i}{4 - 6 i + 6 i - 9 i^{2}}$

Add $- 6 i$ and $6 i$ .

$G = \frac{- 5 - 12 i}{4 + 0 - 9 i^{2}}$

Add $4$ and $0$ .

$G = \frac{- 5 - 12 i}{4 - 9 i^{2}}$

Simplify each term.

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

$G = \frac{- 5 - 12 i}{4 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$G = \frac{- 5 - 12 i}{4 + 9}$

Add $4$ and $9$ .

$G = \frac{- 5 - 12 i}{13}$

Step 3

Split the fraction $\frac{- 5 - 12 i}{13}$ into two fractions.

$G = \frac{- 5}{13} + \frac{- 12 i}{13}$

Step 4

Simplify each term.

Move the negative in front of the fraction.

$G = - \frac{5}{13} + \frac{- 12 i}{13}$

Move the negative in front of the fraction.

$G = - \frac{5}{13} - \frac{12 i}{13}$

Solve for G G=(-2+3i)/(-2-3i)

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