Divide (7-10i)/(-6-6i)

$\frac{7 - 10 i}{- 6 - 6 i}$

Step 1

Multiply the numerator and denominator of $\frac{7 - 10 i}{- 6 - 6 i}$ by the conjugate of $- 6 - 6 i$ to make the denominator real.

$\frac{7 - 10 i}{- 6 - 6 i} \cdot \frac{- 6 + 6 i}{- 6 + 6 i}$

Step 2

Multiply.

Combine.

$\frac{(7 - 10 i) (- 6 + 6 i)}{(- 6 - 6 i) (- 6 + 6 i)}$

Simplify the numerator.

Expand $(7 - 10 i) (- 6 + 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{7 (- 6 + 6 i) - 10 i (- 6 + 6 i)}{(- 6 - 6 i) (- 6 + 6 i)}$

Apply the distributive property.

$\frac{7 \cdot - 6 + 7 (6 i) - 10 i (- 6 + 6 i)}{(- 6 - 6 i) (- 6 + 6 i)}$

Apply the distributive property.

$\frac{7 \cdot - 6 + 7 (6 i) - 10 i \cdot - 6 - 10 i (6 i)}{(- 6 - 6 i) (- 6 + 6 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $- 6$ .

$\frac{- 42 + 7 (6 i) - 10 i \cdot - 6 - 10 i (6 i)}{(- 6 - 6 i) (- 6 + 6 i)}$

Multiply $6$ by $7$ .

$\frac{- 42 + 42 i - 10 i \cdot - 6 - 10 i (6 i)}{(- 6 - 6 i) (- 6 + 6 i)}$

Multiply $- 6$ by $- 10$ .

$\frac{- 42 + 42 i + 60 i - 10 i (6 i)}{(- 6 - 6 i) (- 6 + 6 i)}$

Multiply $- 10 i (6 i)$ .

Multiply $6$ by $- 10$ .

$\frac{- 42 + 42 i + 60 i - 60 i i}{(- 6 - 6 i) (- 6 + 6 i)}$

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

$\frac{- 42 + 42 i + 60 i - 60 (i^{1} i)}{(- 6 - 6 i) (- 6 + 6 i)}$

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

$\frac{- 42 + 42 i + 60 i - 60 (i^{1} i^{1})}{(- 6 - 6 i) (- 6 + 6 i)}$

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

$\frac{- 42 + 42 i + 60 i - 60 i^{1 + 1}}{(- 6 - 6 i) (- 6 + 6 i)}$

Add $1$ and $1$ .

$\frac{- 42 + 42 i + 60 i - 60 i^{2}}{(- 6 - 6 i) (- 6 + 6 i)}$

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

$\frac{- 42 + 42 i + 60 i - 60 \cdot - 1}{(- 6 - 6 i) (- 6 + 6 i)}$

Multiply $- 60$ by $- 1$ .

$\frac{- 42 + 42 i + 60 i + 60}{(- 6 - 6 i) (- 6 + 6 i)}$

Add $- 42$ and $60$ .

$\frac{18 + 42 i + 60 i}{(- 6 - 6 i) (- 6 + 6 i)}$

Add $42 i$ and $60 i$ .

$\frac{18 + 102 i}{(- 6 - 6 i) (- 6 + 6 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{18 + 102 i}{- 6 (- 6 + 6 i) - 6 i (- 6 + 6 i)}$

Apply the distributive property.

$\frac{18 + 102 i}{- 6 \cdot - 6 - 6 (6 i) - 6 i (- 6 + 6 i)}$

Apply the distributive property.

$\frac{18 + 102 i}{- 6 \cdot - 6 - 6 (6 i) - 6 i \cdot - 6 - 6 i (6 i)}$

Simplify.

Multiply $- 6$ by $- 6$ .

$\frac{18 + 102 i}{36 - 6 (6 i) - 6 i \cdot - 6 - 6 i (6 i)}$

Multiply $6$ by $- 6$ .

$\frac{18 + 102 i}{36 - 36 i - 6 i \cdot - 6 - 6 i (6 i)}$

Multiply $- 6$ by $- 6$ .

$\frac{18 + 102 i}{36 - 36 i + 36 i - 6 i (6 i)}$

Multiply $6$ by $- 6$ .

$\frac{18 + 102 i}{36 - 36 i + 36 i - 36 i i}$

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

$\frac{18 + 102 i}{36 - 36 i + 36 i - 36 (i^{1} i)}$

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

$\frac{18 + 102 i}{36 - 36 i + 36 i - 36 (i^{1} i^{1})}$

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

$\frac{18 + 102 i}{36 - 36 i + 36 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{18 + 102 i}{36 - 36 i + 36 i - 36 i^{2}}$

Add $- 36 i$ and $36 i$ .

$\frac{18 + 102 i}{36 + 0 - 36 i^{2}}$

Add $36$ and $0$ .

$\frac{18 + 102 i}{36 - 36 i^{2}}$

Simplify each term.

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

$\frac{18 + 102 i}{36 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{18 + 102 i}{36 + 36}$

Add $36$ and $36$ .

$\frac{18 + 102 i}{72}$

Step 3

Cancel the common factor of $18 + 102 i$ and $72$ .

Factor $6$ out of $18$ .

$\frac{6 (3) + 102 i}{72}$

Factor $6$ out of $102 i$ .

$\frac{6 (3) + 6 (17 i)}{72}$

Factor $6$ out of $6 (3) + 6 (17 i)$ .

$\frac{6 (3 + 17 i)}{72}$

Cancel the common factors.

Factor $6$ out of $72$ .

$\frac{6 (3 + 17 i)}{6 \cdot 12}$

Cancel the common factor.

$\frac{6 (3 + 17 i)}{6 \cdot 12}$

Rewrite the expression.

$\frac{3 + 17 i}{12}$

Step 4

Split the fraction $\frac{3 + 17 i}{12}$ into two fractions.

$\frac{3}{12} + \frac{17 i}{12}$

Step 5

Cancel the common factor of $3$ and $12$ .

Factor $3$ out of $3$ .

$\frac{3 (1)}{12} + \frac{17 i}{12}$

Cancel the common factors.

Factor $3$ out of $12$ .

$\frac{3 \cdot 1}{3 \cdot 4} + \frac{17 i}{12}$

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 4} + \frac{17 i}{12}$

Rewrite the expression.

$\frac{1}{4} + \frac{17 i}{12}$

Divide (7-10i)/(-6-6i)

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