Simplify (10-10i)/(2+4i)

$\frac{10 - 10 i}{2 + 4 i}$

Cancel the common factor of $10 - 10 i$ and $2 + 4 i$ .

Factor $2$ out of $10$ .

$\frac{2 \cdot 5 - 10 i}{2 + 4 i}$

Factor $2$ out of $- 10 i$ .

$\frac{2 \cdot 5 + 2 (- 5 i)}{2 + 4 i}$

Factor $2$ out of $2 (5) + 2 (- 5 i)$ .

$\frac{2 (5 - 5 i)}{2 + 4 i}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 (5 - 5 i)}{2 (1) + 4 i}$

Factor $2$ out of $4 i$ .

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

Factor $2$ out of $2 (1) + 2 (2 i)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $1$ .

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

Multiply $- 2$ by $5$ .

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

Multiply $- 5$ by $1$ .

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

Multiply $- 5 i (- 2 i)$ .

Multiply $- 2$ by $- 5$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $10$ by $- 1$ .

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

Subtract $10$ from $5$ .

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

Subtract $5 i$ from $- 10 i$ .

$\frac{- 5 - 15 i}{(1 + 2 i) (1 - 2 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 5 - 15 i}{1 (1 - 2 i) + 2 i (1 - 2 i)}$

Apply the distributive property.

$\frac{- 5 - 15 i}{1 \cdot 1 + 1 (- 2 i) + 2 i (1 - 2 i)}$

Apply the distributive property.

$\frac{- 5 - 15 i}{1 \cdot 1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)}$

Simplify.

Multiply $1$ by $1$ .

$\frac{- 5 - 15 i}{1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)}$

Multiply $- 2$ by $1$ .

$\frac{- 5 - 15 i}{1 - 2 i + 2 i \cdot 1 + 2 i (- 2 i)}$

Multiply $2$ by $1$ .

$\frac{- 5 - 15 i}{1 - 2 i + 2 i + 2 i (- 2 i)}$

Multiply $- 2$ by $2$ .

$\frac{- 5 - 15 i}{1 - 2 i + 2 i - 4 i i}$

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

$\frac{- 5 - 15 i}{1 - 2 i + 2 i - 4 (i^{1} i)}$

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

$\frac{- 5 - 15 i}{1 - 2 i + 2 i - 4 (i^{1} i^{1})}$

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

$\frac{- 5 - 15 i}{1 - 2 i + 2 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 5 - 15 i}{1 - 2 i + 2 i - 4 i^{2}}$

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

$\frac{- 5 - 15 i}{1 + 0 - 4 i^{2}}$

Add $1$ and $0$ .

$\frac{- 5 - 15 i}{1 - 4 i^{2}}$

Simplify each term.

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

$\frac{- 5 - 15 i}{1 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{- 5 - 15 i}{1 + 4}$

Add $1$ and $4$ .

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

Cancel the common factor of $- 5 - 15 i$ and $5$ .

Factor $5$ out of $- 5$ .

$\frac{5 \cdot - 1 - 15 i}{5}$

Factor $5$ out of $- 15 i$ .

$\frac{5 \cdot - 1 + 5 (- 3 i)}{5}$

Factor $5$ out of $5 (- 1) + 5 (- 3 i)$ .

$\frac{5 (- 1 - 3 i)}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{5 (- 1 - 3 i)}{5 (1)}$

Cancel the common factor.

$\frac{5 (- 1 - 3 i)}{5 \cdot 1}$

Rewrite the expression.

$\frac{- 1 - 3 i}{1}$

Divide $- 1 - 3 i$ by $1$ .

$- 1 - 3 i$

Simplify (10-10i)/(2+4i)

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