Simplify (-9+8i)/(-3+9i)

$\frac{- 9 + 8 i}{- 3 + 9 i}$

Multiply the numerator and denominator of $\frac{- 9 + 8 i}{- 3 + 9 i}$ by the conjugate of $- 3 + 9 i$ to make the denominator real.

$\frac{- 9 + 8 i}{- 3 + 9 i} \cdot \frac{- 3 - 9 i}{- 3 - 9 i}$

Multiply.

Combine.

$\frac{(- 9 + 8 i) (- 3 - 9 i)}{(- 3 + 9 i) (- 3 - 9 i)}$

Simplify the numerator.

Expand $(- 9 + 8 i) (- 3 - 9 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 9 (- 3 - 9 i) + 8 i (- 3 - 9 i)}{(- 3 + 9 i) (- 3 - 9 i)}$

Apply the distributive property.

$\frac{- 9 \cdot - 3 - 9 (- 9 i) + 8 i (- 3 - 9 i)}{(- 3 + 9 i) (- 3 - 9 i)}$

Apply the distributive property.

$\frac{- 9 \cdot - 3 - 9 (- 9 i) + 8 i \cdot - 3 + 8 i (- 9 i)}{(- 3 + 9 i) (- 3 - 9 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 9$ by $- 3$ .

$\frac{27 - 9 (- 9 i) + 8 i \cdot - 3 + 8 i (- 9 i)}{(- 3 + 9 i) (- 3 - 9 i)}$

Multiply $- 9$ by $- 9$ .

$\frac{27 + 81 i + 8 i \cdot - 3 + 8 i (- 9 i)}{(- 3 + 9 i) (- 3 - 9 i)}$

Multiply $- 3$ by $8$ .

$\frac{27 + 81 i - 24 i + 8 i (- 9 i)}{(- 3 + 9 i) (- 3 - 9 i)}$

Multiply $8 i (- 9 i)$ .

Multiply $- 9$ by $8$ .

$\frac{27 + 81 i - 24 i - 72 i i}{(- 3 + 9 i) (- 3 - 9 i)}$

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

$\frac{27 + 81 i - 24 i - 72 (i^{1} i)}{(- 3 + 9 i) (- 3 - 9 i)}$

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

$\frac{27 + 81 i - 24 i - 72 (i^{1} i^{1})}{(- 3 + 9 i) (- 3 - 9 i)}$

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

$\frac{27 + 81 i - 24 i - 72 i^{1 + 1}}{(- 3 + 9 i) (- 3 - 9 i)}$

Add $1$ and $1$ .

$\frac{27 + 81 i - 24 i - 72 i^{2}}{(- 3 + 9 i) (- 3 - 9 i)}$

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

$\frac{27 + 81 i - 24 i - 72 \cdot - 1}{(- 3 + 9 i) (- 3 - 9 i)}$

Multiply $- 72$ by $- 1$ .

$\frac{27 + 81 i - 24 i + 72}{(- 3 + 9 i) (- 3 - 9 i)}$

Add $27$ and $72$ .

$\frac{99 + 81 i - 24 i}{(- 3 + 9 i) (- 3 - 9 i)}$

Subtract $24 i$ from $81 i$ .

$\frac{99 + 57 i}{(- 3 + 9 i) (- 3 - 9 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{99 + 57 i}{- 3 (- 3 - 9 i) + 9 i (- 3 - 9 i)}$

Apply the distributive property.

$\frac{99 + 57 i}{- 3 \cdot - 3 - 3 (- 9 i) + 9 i (- 3 - 9 i)}$

Apply the distributive property.

$\frac{99 + 57 i}{- 3 \cdot - 3 - 3 (- 9 i) + 9 i \cdot - 3 + 9 i (- 9 i)}$

Simplify.

Multiply $- 3$ by $- 3$ .

$\frac{99 + 57 i}{9 - 3 (- 9 i) + 9 i \cdot - 3 + 9 i (- 9 i)}$

Multiply $- 9$ by $- 3$ .

$\frac{99 + 57 i}{9 + 27 i + 9 i \cdot - 3 + 9 i (- 9 i)}$

Multiply $- 3$ by $9$ .

$\frac{99 + 57 i}{9 + 27 i - 27 i + 9 i (- 9 i)}$

Multiply $- 9$ by $9$ .

$\frac{99 + 57 i}{9 + 27 i - 27 i - 81 i i}$

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

$\frac{99 + 57 i}{9 + 27 i - 27 i - 81 (i^{1} i)}$

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

$\frac{99 + 57 i}{9 + 27 i - 27 i - 81 (i^{1} i^{1})}$

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

$\frac{99 + 57 i}{9 + 27 i - 27 i - 81 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{99 + 57 i}{9 + 27 i - 27 i - 81 i^{2}}$

Subtract $27 i$ from $27 i$ .

$\frac{99 + 57 i}{9 + 0 - 81 i^{2}}$

Add $9$ and $0$ .

$\frac{99 + 57 i}{9 - 81 i^{2}}$

Simplify each term.

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

$\frac{99 + 57 i}{9 - 81 \cdot - 1}$

Multiply $- 81$ by $- 1$ .

$\frac{99 + 57 i}{9 + 81}$

Add $9$ and $81$ .

$\frac{99 + 57 i}{90}$

Cancel the common factor of $99 + 57 i$ and $90$ .

Factor $3$ out of $99$ .

$\frac{3 (33) + 57 i}{90}$

Factor $3$ out of $57 i$ .

$\frac{3 (33) + 3 (19 i)}{90}$

Factor $3$ out of $3 (33) + 3 (19 i)$ .

$\frac{3 (33 + 19 i)}{90}$

Cancel the common factors.

Factor $3$ out of $90$ .

$\frac{3 (33 + 19 i)}{3 \cdot 30}$

Cancel the common factor.

$\frac{3 (33 + 19 i)}{3 \cdot 30}$

Rewrite the expression.

$\frac{33 + 19 i}{30}$

Split the fraction $\frac{33 + 19 i}{30}$ into two fractions.

$\frac{33}{30} + \frac{19 i}{30}$

Cancel the common factor of $33$ and $30$ .

Factor $3$ out of $33$ .

$\frac{3 (11)}{30} + \frac{19 i}{30}$

Cancel the common factors.

Factor $3$ out of $30$ .

$\frac{3 \cdot 11}{3 \cdot 10} + \frac{19 i}{30}$

Cancel the common factor.

$\frac{3 \cdot 11}{3 \cdot 10} + \frac{19 i}{30}$

Rewrite the expression.

$\frac{11}{10} + \frac{19 i}{30}$

Simplify (-9+8i)/(-3+9i)

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