Simplify (2-7i)/(3+6i)

$\frac{(2 - 7 i)}{(3 + 6 i)}$

Multiply the numerator and denominator of $\frac{2 - 7 i}{3 + 6 i}$ by the conjugate of $3 + 6 i$ to make the denominator real.

$\frac{2 - 7 i}{3 + 6 i} \cdot \frac{3 - 6 i}{3 - 6 i}$

Multiply.

Combine.

$\frac{(2 - 7 i) (3 - 6 i)}{(3 + 6 i) (3 - 6 i)}$

Simplify the numerator.

Expand $(2 - 7 i) (3 - 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (3 - 6 i) - 7 i (3 - 6 i)}{(3 + 6 i) (3 - 6 i)}$

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- 6 i) - 7 i (3 - 6 i)}{(3 + 6 i) (3 - 6 i)}$

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- 6 i) - 7 i \cdot 3 - 7 i (- 6 i)}{(3 + 6 i) (3 - 6 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $3$ .

$\frac{6 + 2 (- 6 i) - 7 i \cdot 3 - 7 i (- 6 i)}{(3 + 6 i) (3 - 6 i)}$

Multiply $- 6$ by $2$ .

$\frac{6 - 12 i - 7 i \cdot 3 - 7 i (- 6 i)}{(3 + 6 i) (3 - 6 i)}$

Multiply $3$ by $- 7$ .

$\frac{6 - 12 i - 21 i - 7 i (- 6 i)}{(3 + 6 i) (3 - 6 i)}$

Multiply $- 7 i (- 6 i)$ .

Multiply $- 6$ by $- 7$ .

$\frac{6 - 12 i - 21 i + 42 i i}{(3 + 6 i) (3 - 6 i)}$

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

$\frac{6 - 12 i - 21 i + 42 (i^{1} i)}{(3 + 6 i) (3 - 6 i)}$

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

$\frac{6 - 12 i - 21 i + 42 (i^{1} i^{1})}{(3 + 6 i) (3 - 6 i)}$

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

$\frac{6 - 12 i - 21 i + 42 i^{1 + 1}}{(3 + 6 i) (3 - 6 i)}$

Add $1$ and $1$ .

$\frac{6 - 12 i - 21 i + 42 i^{2}}{(3 + 6 i) (3 - 6 i)}$

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

$\frac{6 - 12 i - 21 i + 42 \cdot - 1}{(3 + 6 i) (3 - 6 i)}$

Multiply $42$ by $- 1$ .

$\frac{6 - 12 i - 21 i - 42}{(3 + 6 i) (3 - 6 i)}$

Subtract $42$ from $6$ .

$\frac{- 36 - 12 i - 21 i}{(3 + 6 i) (3 - 6 i)}$

Subtract $21 i$ from $- 12 i$ .

$\frac{- 36 - 33 i}{(3 + 6 i) (3 - 6 i)}$

Simplify the denominator.

Expand $(3 + 6 i) (3 - 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 36 - 33 i}{3 (3 - 6 i) + 6 i (3 - 6 i)}$

Apply the distributive property.

$\frac{- 36 - 33 i}{3 \cdot 3 + 3 (- 6 i) + 6 i (3 - 6 i)}$

Apply the distributive property.

$\frac{- 36 - 33 i}{3 \cdot 3 + 3 (- 6 i) + 6 i \cdot 3 + 6 i (- 6 i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{- 36 - 33 i}{9 + 3 (- 6 i) + 6 i \cdot 3 + 6 i (- 6 i)}$

Multiply $- 6$ by $3$ .

$\frac{- 36 - 33 i}{9 - 18 i + 6 i \cdot 3 + 6 i (- 6 i)}$

Multiply $3$ by $6$ .

$\frac{- 36 - 33 i}{9 - 18 i + 18 i + 6 i (- 6 i)}$

Multiply $- 6$ by $6$ .

$\frac{- 36 - 33 i}{9 - 18 i + 18 i - 36 i i}$

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

$\frac{- 36 - 33 i}{9 - 18 i + 18 i - 36 (i^{1} i)}$

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

$\frac{- 36 - 33 i}{9 - 18 i + 18 i - 36 (i^{1} i^{1})}$

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

$\frac{- 36 - 33 i}{9 - 18 i + 18 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 36 - 33 i}{9 - 18 i + 18 i - 36 i^{2}}$

Add $- 18 i$ and $18 i$ .

$\frac{- 36 - 33 i}{9 + 0 - 36 i^{2}}$

Add $9$ and $0$ .

$\frac{- 36 - 33 i}{9 - 36 i^{2}}$

Simplify each term.

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

$\frac{- 36 - 33 i}{9 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{- 36 - 33 i}{9 + 36}$

Add $9$ and $36$ .

$\frac{- 36 - 33 i}{45}$

Cancel the common factor of $- 36 - 33 i$ and $45$ .

Factor $3$ out of $- 36$ .

$\frac{3 (- 12) - 33 i}{45}$

Factor $3$ out of $- 33 i$ .

$\frac{3 (- 12) + 3 (- 11 i)}{45}$

Factor $3$ out of $3 (- 12) + 3 (- 11 i)$ .

$\frac{3 (- 12 - 11 i)}{45}$

Cancel the common factors.

Factor $3$ out of $45$ .

$\frac{3 (- 12 - 11 i)}{3 \cdot 15}$

Cancel the common factor.

$\frac{3 (- 12 - 11 i)}{3 \cdot 15}$

Rewrite the expression.

$\frac{- 12 - 11 i}{15}$

Split the fraction $\frac{- 12 - 11 i}{15}$ into two fractions.

$\frac{- 12}{15} + \frac{- 11 i}{15}$

Simplify each term.

Cancel the common factor of $- 12$ and $15$ .

Factor $3$ out of $- 12$ .

$\frac{3 (- 4)}{15} + \frac{- 11 i}{15}$

Cancel the common factors.

Factor $3$ out of $15$ .

$\frac{3 \cdot - 4}{3 \cdot 5} + \frac{- 11 i}{15}$

Cancel the common factor.

$\frac{3 \cdot - 4}{3 \cdot 5} + \frac{- 11 i}{15}$

Rewrite the expression.

$\frac{- 4}{5} + \frac{- 11 i}{15}$

Move the negative in front of the fraction.

$- \frac{4}{5} + \frac{- 11 i}{15}$

Move the negative in front of the fraction.

$- \frac{4}{5} - \frac{11 i}{15}$

Simplify (2-7i)/(3+6i)

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