Simplify (-66+27i)/(-6-3i)

$\frac{- 66 + 27 i}{- 6 - 3 i}$

Step 1

Cancel the common factor of $- 66 + 27 i$ and $- 6 - 3 i$ .

Factor $3$ out of $- 66$ .

$\frac{3 (- 22) + 27 i}{- 6 - 3 i}$

Factor $3$ out of $27 i$ .

$\frac{3 (- 22) + 3 (9 i)}{- 6 - 3 i}$

Factor $3$ out of $3 (- 22) + 3 (9 i)$ .

$\frac{3 (- 22 + 9 i)}{- 6 - 3 i}$

Cancel the common factors.

Factor $3$ out of $- 6$ .

$\frac{3 (- 22 + 9 i)}{3 \cdot - 2 - 3 i}$

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

$\frac{3 (- 22 + 9 i)}{3 \cdot - 2 + 3 (- i)}$

Factor $3$ out of $3 (- 2) + 3 (- i)$ .

$\frac{3 (- 22 + 9 i)}{3 (- 2 - i)}$

Cancel the common factor.

$\frac{3 (- 22 + 9 i)}{3 (- 2 - i)}$

Rewrite the expression.

$\frac{- 22 + 9 i}{- 2 - i}$

Step 2

Multiply the numerator and denominator of $\frac{- 22 + 9 i}{- 2 - i}$ by the conjugate of $- 2 - i$ to make the denominator real.

$\frac{- 22 + 9 i}{- 2 - i} \cdot \frac{- 2 + i}{- 2 + i}$

Step 3

Multiply.

Combine.

$\frac{(- 22 + 9 i) (- 2 + i)}{(- 2 - i) (- 2 + i)}$

Simplify the numerator.

Expand $(- 22 + 9 i) (- 2 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 22 (- 2 + i) + 9 i (- 2 + i)}{(- 2 - i) (- 2 + i)}$

Apply the distributive property.

$\frac{- 22 \cdot - 2 - 22 i + 9 i (- 2 + i)}{(- 2 - i) (- 2 + i)}$

Apply the distributive property.

$\frac{- 22 \cdot - 2 - 22 i + 9 i \cdot - 2 + 9 i i}{(- 2 - i) (- 2 + i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 22$ by $- 2$ .

$\frac{44 - 22 i + 9 i \cdot - 2 + 9 i i}{(- 2 - i) (- 2 + i)}$

Multiply $- 2$ by $9$ .

$\frac{44 - 22 i - 18 i + 9 i i}{(- 2 - i) (- 2 + i)}$

Multiply $9 i i$ .

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

$\frac{44 - 22 i - 18 i + 9 (i^{1} i)}{(- 2 - i) (- 2 + i)}$

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

$\frac{44 - 22 i - 18 i + 9 (i^{1} i^{1})}{(- 2 - i) (- 2 + i)}$

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

$\frac{44 - 22 i - 18 i + 9 i^{1 + 1}}{(- 2 - i) (- 2 + i)}$

Add $1$ and $1$ .

$\frac{44 - 22 i - 18 i + 9 i^{2}}{(- 2 - i) (- 2 + i)}$

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

$\frac{44 - 22 i - 18 i + 9 \cdot - 1}{(- 2 - i) (- 2 + i)}$

Multiply $9$ by $- 1$ .

$\frac{44 - 22 i - 18 i - 9}{(- 2 - i) (- 2 + i)}$

Subtract $9$ from $44$ .

$\frac{35 - 22 i - 18 i}{(- 2 - i) (- 2 + i)}$

Subtract $18 i$ from $- 22 i$ .

$\frac{35 - 40 i}{(- 2 - i) (- 2 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{35 - 40 i}{- 2 (- 2 + i) - i (- 2 + i)}$

Apply the distributive property.

$\frac{35 - 40 i}{- 2 \cdot - 2 - 2 i - i (- 2 + i)}$

Apply the distributive property.

$\frac{35 - 40 i}{- 2 \cdot - 2 - 2 i - i \cdot - 2 - i i}$

Simplify.

Multiply $- 2$ by $- 2$ .

$\frac{35 - 40 i}{4 - 2 i - i \cdot - 2 - i i}$

Multiply $- 2$ by $- 1$ .

$\frac{35 - 40 i}{4 - 2 i + 2 i - i i}$

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

$\frac{35 - 40 i}{4 - 2 i + 2 i - (i^{1} i)}$

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

$\frac{35 - 40 i}{4 - 2 i + 2 i - (i^{1} i^{1})}$

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

$\frac{35 - 40 i}{4 - 2 i + 2 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{35 - 40 i}{4 - 2 i + 2 i - i^{2}}$

Add $- 2 i$ and $2 i$ .

$\frac{35 - 40 i}{4 + 0 - i^{2}}$

Add $4$ and $0$ .

$\frac{35 - 40 i}{4 - i^{2}}$

Simplify each term.

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

$\frac{35 - 40 i}{4 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{35 - 40 i}{4 + 1}$

Add $4$ and $1$ .

$\frac{35 - 40 i}{5}$

Step 4

Cancel the common factor of $35 - 40 i$ and $5$ .

Factor $5$ out of $35$ .

$\frac{5 \cdot 7 - 40 i}{5}$

Factor $5$ out of $- 40 i$ .

$\frac{5 \cdot 7 + 5 (- 8 i)}{5}$

Factor $5$ out of $5 (7) + 5 (- 8 i)$ .

$\frac{5 (7 - 8 i)}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{5 (7 - 8 i)}{5 (1)}$

Cancel the common factor.

$\frac{5 (7 - 8 i)}{5 \cdot 1}$

Rewrite the expression.

$\frac{7 - 8 i}{1}$

Divide $7 - 8 i$ by $1$ .

$7 - 8 i$

Simplify (-66+27i)/(-6-3i)

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