Simplify (-10+5i)/(-2i-1)

$\frac{- 10 + 5 i}{- 2 i - 1}$

Step 1

Multiply the numerator and denominator of $\frac{- 10 + 5 i}{- 1 - 2 i}$ by the conjugate of $- 1 - 2 i$ to make the denominator real.

$\frac{- 10 + 5 i}{- 1 - 2 i} \cdot \frac{- 1 + 2 i}{- 1 + 2 i}$

Step 2

Multiply.

Combine.

$\frac{(- 10 + 5 i) (- 1 + 2 i)}{(- 1 - 2 i) (- 1 + 2 i)}$

Simplify the numerator.

Expand $(- 10 + 5 i) (- 1 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 10 (- 1 + 2 i) + 5 i (- 1 + 2 i)}{(- 1 - 2 i) (- 1 + 2 i)}$

Apply the distributive property.

$\frac{- 10 \cdot - 1 - 10 (2 i) + 5 i (- 1 + 2 i)}{(- 1 - 2 i) (- 1 + 2 i)}$

Apply the distributive property.

$\frac{- 10 \cdot - 1 - 10 (2 i) + 5 i \cdot - 1 + 5 i (2 i)}{(- 1 - 2 i) (- 1 + 2 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 10$ by $- 1$ .

$\frac{10 - 10 (2 i) + 5 i \cdot - 1 + 5 i (2 i)}{(- 1 - 2 i) (- 1 + 2 i)}$

Multiply $2$ by $- 10$ .

$\frac{10 - 20 i + 5 i \cdot - 1 + 5 i (2 i)}{(- 1 - 2 i) (- 1 + 2 i)}$

Multiply $- 1$ by $5$ .

$\frac{10 - 20 i - 5 i + 5 i (2 i)}{(- 1 - 2 i) (- 1 + 2 i)}$

Multiply $5 i (2 i)$ .

Multiply $2$ by $5$ .

$\frac{10 - 20 i - 5 i + 10 i i}{(- 1 - 2 i) (- 1 + 2 i)}$

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

$\frac{10 - 20 i - 5 i + 10 (i^{1} i)}{(- 1 - 2 i) (- 1 + 2 i)}$

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

$\frac{10 - 20 i - 5 i + 10 (i^{1} i^{1})}{(- 1 - 2 i) (- 1 + 2 i)}$

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

$\frac{10 - 20 i - 5 i + 10 i^{1 + 1}}{(- 1 - 2 i) (- 1 + 2 i)}$

Add $1$ and $1$ .

$\frac{10 - 20 i - 5 i + 10 i^{2}}{(- 1 - 2 i) (- 1 + 2 i)}$

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

$\frac{10 - 20 i - 5 i + 10 \cdot - 1}{(- 1 - 2 i) (- 1 + 2 i)}$

Multiply $10$ by $- 1$ .

$\frac{10 - 20 i - 5 i - 10}{(- 1 - 2 i) (- 1 + 2 i)}$

Subtract $10$ from $10$ .

$\frac{0 - 20 i - 5 i}{(- 1 - 2 i) (- 1 + 2 i)}$

Subtract $20 i$ from $0$ .

$\frac{- 20 i - 5 i}{(- 1 - 2 i) (- 1 + 2 i)}$

Subtract $5 i$ from $- 20 i$ .

$\frac{- 25 i}{(- 1 - 2 i) (- 1 + 2 i)}$

Simplify the denominator.

Expand $(- 1 - 2 i) (- 1 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 25 i}{- 1 (- 1 + 2 i) - 2 i (- 1 + 2 i)}$

Apply the distributive property.

$\frac{- 25 i}{- 1 \cdot - 1 - 1 (2 i) - 2 i (- 1 + 2 i)}$

Apply the distributive property.

$\frac{- 25 i}{- 1 \cdot - 1 - 1 (2 i) - 2 i \cdot - 1 - 2 i (2 i)}$

Simplify.

Multiply $- 1$ by $- 1$ .

$\frac{- 25 i}{1 - 1 (2 i) - 2 i \cdot - 1 - 2 i (2 i)}$

Multiply $2$ by $- 1$ .

$\frac{- 25 i}{1 - 2 i - 2 i \cdot - 1 - 2 i (2 i)}$

Multiply $- 1$ by $- 2$ .

$\frac{- 25 i}{1 - 2 i + 2 i - 2 i (2 i)}$

Multiply $2$ by $- 2$ .

$\frac{- 25 i}{1 - 2 i + 2 i - 4 i i}$

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

$\frac{- 25 i}{1 - 2 i + 2 i - 4 (i^{1} i)}$

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

$\frac{- 25 i}{1 - 2 i + 2 i - 4 (i^{1} i^{1})}$

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

$\frac{- 25 i}{1 - 2 i + 2 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 25 i}{1 - 2 i + 2 i - 4 i^{2}}$

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

$\frac{- 25 i}{1 + 0 - 4 i^{2}}$

Add $1$ and $0$ .

$\frac{- 25 i}{1 - 4 i^{2}}$

Simplify each term.

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

$\frac{- 25 i}{1 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{- 25 i}{1 + 4}$

Add $1$ and $4$ .

$\frac{- 25 i}{5}$

Step 3

Cancel the common factor of $- 25$ and $5$ .

Factor $5$ out of $- 25 i$ .

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

Cancel the common factors.

Factor $5$ out of $5$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 5 i}{1}$

Divide $- 5 i$ by $1$ .

$- 5 i$

Simplify (-10+5i)/(-2i-1)

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