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

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

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

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

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 2$ by $5$ .

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

Multiply $5 i (6 i)$ .

Multiply $6$ by $5$ .

$\frac{- 10 i + 30 i i}{(- 2 - 6 i) (- 2 + 6 i)}$

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

$\frac{- 10 i + 30 (i^{1} i)}{(- 2 - 6 i) (- 2 + 6 i)}$

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

$\frac{- 10 i + 30 (i^{1} i^{1})}{(- 2 - 6 i) (- 2 + 6 i)}$

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

$\frac{- 10 i + 30 i^{1 + 1}}{(- 2 - 6 i) (- 2 + 6 i)}$

Add $1$ and $1$ .

$\frac{- 10 i + 30 i^{2}}{(- 2 - 6 i) (- 2 + 6 i)}$

Simplify each term.

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

$\frac{- 10 i + 30 \cdot - 1}{(- 2 - 6 i) (- 2 + 6 i)}$

Multiply $30$ by $- 1$ .

$\frac{- 10 i - 30}{(- 2 - 6 i) (- 2 + 6 i)}$

Reorder $- 10 i$ and $- 30$ .

$\frac{- 30 - 10 i}{(- 2 - 6 i) (- 2 + 6 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 30 - 10 i}{- 2 (- 2 + 6 i) - 6 i (- 2 + 6 i)}$

Apply the distributive property.

$\frac{- 30 - 10 i}{- 2 \cdot - 2 - 2 (6 i) - 6 i (- 2 + 6 i)}$

Apply the distributive property.

$\frac{- 30 - 10 i}{- 2 \cdot - 2 - 2 (6 i) - 6 i \cdot - 2 - 6 i (6 i)}$

Simplify.

Multiply $- 2$ by $- 2$ .

$\frac{- 30 - 10 i}{4 - 2 (6 i) - 6 i \cdot - 2 - 6 i (6 i)}$

Multiply $6$ by $- 2$ .

$\frac{- 30 - 10 i}{4 - 12 i - 6 i \cdot - 2 - 6 i (6 i)}$

Multiply $- 2$ by $- 6$ .

$\frac{- 30 - 10 i}{4 - 12 i + 12 i - 6 i (6 i)}$

Multiply $6$ by $- 6$ .

$\frac{- 30 - 10 i}{4 - 12 i + 12 i - 36 i i}$

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

$\frac{- 30 - 10 i}{4 - 12 i + 12 i - 36 (i^{1} i)}$

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

$\frac{- 30 - 10 i}{4 - 12 i + 12 i - 36 (i^{1} i^{1})}$

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

$\frac{- 30 - 10 i}{4 - 12 i + 12 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 30 - 10 i}{4 - 12 i + 12 i - 36 i^{2}}$

Add $- 12 i$ and $12 i$ .

$\frac{- 30 - 10 i}{4 + 0 - 36 i^{2}}$

Add $4$ and $0$ .

$\frac{- 30 - 10 i}{4 - 36 i^{2}}$

Simplify each term.

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

$\frac{- 30 - 10 i}{4 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{- 30 - 10 i}{4 + 36}$

Add $4$ and $36$ .

$\frac{- 30 - 10 i}{40}$

Cancel the common factor of $- 30 - 10 i$ and $40$ .

Factor $10$ out of $- 30$ .

$\frac{10 (- 3) - 10 i}{40}$

Factor $10$ out of $- 10 i$ .

$\frac{10 (- 3) + 10 (- i)}{40}$

Factor $10$ out of $10 (- 3) + 10 (- i)$ .

$\frac{10 (- 3 - i)}{40}$

Cancel the common factors.

Factor $10$ out of $40$ .

$\frac{10 (- 3 - i)}{10 \cdot 4}$

Cancel the common factor.

$\frac{10 (- 3 - i)}{10 \cdot 4}$

Rewrite the expression.

$\frac{- 3 - i}{4}$

Split the fraction $\frac{- 3 - i}{4}$ into two fractions.

$\frac{- 3}{4} + \frac{- i}{4}$

Simplify each term.

Move the negative in front of the fraction.

$- \frac{3}{4} + \frac{- i}{4}$

Move the negative in front of the fraction.

$- \frac{3}{4} - \frac{i}{4}$

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

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