Simplify (5+6i)/(-2-8i)

$\frac{5 + 6 i}{- 2 - 8 i}$

Step 1

Multiply the numerator and denominator of $\frac{5 + 6 i}{- 2 - 8 i}$ by the conjugate of $- 2 - 8 i$ to make the denominator real.

$\frac{5 + 6 i}{- 2 - 8 i} \cdot \frac{- 2 + 8 i}{- 2 + 8 i}$

Step 2

Multiply.

Combine.

$\frac{(5 + 6 i) (- 2 + 8 i)}{(- 2 - 8 i) (- 2 + 8 i)}$

Simplify the numerator.

Expand $(5 + 6 i) (- 2 + 8 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 (- 2 + 8 i) + 6 i (- 2 + 8 i)}{(- 2 - 8 i) (- 2 + 8 i)}$

Apply the distributive property.

$\frac{5 \cdot - 2 + 5 (8 i) + 6 i (- 2 + 8 i)}{(- 2 - 8 i) (- 2 + 8 i)}$

Apply the distributive property.

$\frac{5 \cdot - 2 + 5 (8 i) + 6 i \cdot - 2 + 6 i (8 i)}{(- 2 - 8 i) (- 2 + 8 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $- 2$ .

$\frac{- 10 + 5 (8 i) + 6 i \cdot - 2 + 6 i (8 i)}{(- 2 - 8 i) (- 2 + 8 i)}$

Multiply $8$ by $5$ .

$\frac{- 10 + 40 i + 6 i \cdot - 2 + 6 i (8 i)}{(- 2 - 8 i) (- 2 + 8 i)}$

Multiply $- 2$ by $6$ .

$\frac{- 10 + 40 i - 12 i + 6 i (8 i)}{(- 2 - 8 i) (- 2 + 8 i)}$

Multiply $6 i (8 i)$ .

Multiply $8$ by $6$ .

$\frac{- 10 + 40 i - 12 i + 48 i i}{(- 2 - 8 i) (- 2 + 8 i)}$

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

$\frac{- 10 + 40 i - 12 i + 48 (i^{1} i)}{(- 2 - 8 i) (- 2 + 8 i)}$

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

$\frac{- 10 + 40 i - 12 i + 48 (i^{1} i^{1})}{(- 2 - 8 i) (- 2 + 8 i)}$

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

$\frac{- 10 + 40 i - 12 i + 48 i^{1 + 1}}{(- 2 - 8 i) (- 2 + 8 i)}$

Add $1$ and $1$ .

$\frac{- 10 + 40 i - 12 i + 48 i^{2}}{(- 2 - 8 i) (- 2 + 8 i)}$

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

$\frac{- 10 + 40 i - 12 i + 48 \cdot - 1}{(- 2 - 8 i) (- 2 + 8 i)}$

Multiply $48$ by $- 1$ .

$\frac{- 10 + 40 i - 12 i - 48}{(- 2 - 8 i) (- 2 + 8 i)}$

Subtract $48$ from $- 10$ .

$\frac{- 58 + 40 i - 12 i}{(- 2 - 8 i) (- 2 + 8 i)}$

Subtract $12 i$ from $40 i$ .

$\frac{- 58 + 28 i}{(- 2 - 8 i) (- 2 + 8 i)}$

Simplify the denominator.

Expand $(- 2 - 8 i) (- 2 + 8 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 58 + 28 i}{- 2 (- 2 + 8 i) - 8 i (- 2 + 8 i)}$

Apply the distributive property.

$\frac{- 58 + 28 i}{- 2 \cdot - 2 - 2 (8 i) - 8 i (- 2 + 8 i)}$

Apply the distributive property.

$\frac{- 58 + 28 i}{- 2 \cdot - 2 - 2 (8 i) - 8 i \cdot - 2 - 8 i (8 i)}$

Simplify.

Multiply $- 2$ by $- 2$ .

$\frac{- 58 + 28 i}{4 - 2 (8 i) - 8 i \cdot - 2 - 8 i (8 i)}$

Multiply $8$ by $- 2$ .

$\frac{- 58 + 28 i}{4 - 16 i - 8 i \cdot - 2 - 8 i (8 i)}$

Multiply $- 2$ by $- 8$ .

$\frac{- 58 + 28 i}{4 - 16 i + 16 i - 8 i (8 i)}$

Multiply $8$ by $- 8$ .

$\frac{- 58 + 28 i}{4 - 16 i + 16 i - 64 i i}$

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

$\frac{- 58 + 28 i}{4 - 16 i + 16 i - 64 (i^{1} i)}$

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

$\frac{- 58 + 28 i}{4 - 16 i + 16 i - 64 (i^{1} i^{1})}$

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

$\frac{- 58 + 28 i}{4 - 16 i + 16 i - 64 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 58 + 28 i}{4 - 16 i + 16 i - 64 i^{2}}$

Add $- 16 i$ and $16 i$ .

$\frac{- 58 + 28 i}{4 + 0 - 64 i^{2}}$

Add $4$ and $0$ .

$\frac{- 58 + 28 i}{4 - 64 i^{2}}$

Simplify each term.

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

$\frac{- 58 + 28 i}{4 - 64 \cdot - 1}$

Multiply $- 64$ by $- 1$ .

$\frac{- 58 + 28 i}{4 + 64}$

Add $4$ and $64$ .

$\frac{- 58 + 28 i}{68}$

Step 3

Cancel the common factor of $- 58 + 28 i$ and $68$ .

Factor $2$ out of $- 58$ .

$\frac{2 (- 29) + 28 i}{68}$

Factor $2$ out of $28 i$ .

$\frac{2 (- 29) + 2 (14 i)}{68}$

Factor $2$ out of $2 (- 29) + 2 (14 i)$ .

$\frac{2 (- 29 + 14 i)}{68}$

Cancel the common factors.

Factor $2$ out of $68$ .

$\frac{2 (- 29 + 14 i)}{2 \cdot 34}$

Cancel the common factor.

$\frac{2 (- 29 + 14 i)}{2 \cdot 34}$

Rewrite the expression.

$\frac{- 29 + 14 i}{34}$

Step 4

Split the fraction $\frac{- 29 + 14 i}{34}$ into two fractions.

$\frac{- 29}{34} + \frac{14 i}{34}$

Step 5

Simplify each term.

Move the negative in front of the fraction.

$- \frac{29}{34} + \frac{14 i}{34}$

Cancel the common factor of $14$ and $34$ .

Factor $2$ out of $14 i$ .

$- \frac{29}{34} + \frac{2 (7 i)}{34}$

Cancel the common factors.

Factor $2$ out of $34$ .

$- \frac{29}{34} + \frac{2 (7 i)}{2 (17)}$

Cancel the common factor.

$- \frac{29}{34} + \frac{2 (7 i)}{2 \cdot 17}$

Rewrite the expression.

$- \frac{29}{34} + \frac{7 i}{17}$

Simplify (5+6i)/(-2-8i)

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