Simplify (2-4i)/(1+3i)

$\frac{2 - 4 i}{1 + 3 i}$

Multiply the numerator and denominator of $\frac{2 - 4 i}{1 + 3 i}$ by the conjugate of $1 + 3 i$ to make the denominator real.

$\frac{2 - 4 i}{1 + 3 i} \cdot \frac{1 - 3 i}{1 - 3 i}$

Multiply.

Combine.

$\frac{(2 - 4 i) (1 - 3 i)}{(1 + 3 i) (1 - 3 i)}$

Simplify the numerator.

Expand $(2 - 4 i) (1 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (1 - 3 i) - 4 i (1 - 3 i)}{(1 + 3 i) (1 - 3 i)}$

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- 3 i) - 4 i (1 - 3 i)}{(1 + 3 i) (1 - 3 i)}$

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- 3 i) - 4 i \cdot 1 - 4 i (- 3 i)}{(1 + 3 i) (1 - 3 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

$\frac{2 + 2 (- 3 i) - 4 i \cdot 1 - 4 i (- 3 i)}{(1 + 3 i) (1 - 3 i)}$

Multiply $- 3$ by $2$ .

$\frac{2 - 6 i - 4 i \cdot 1 - 4 i (- 3 i)}{(1 + 3 i) (1 - 3 i)}$

Multiply $- 4$ by $1$ .

$\frac{2 - 6 i - 4 i - 4 i (- 3 i)}{(1 + 3 i) (1 - 3 i)}$

Multiply $- 4 i (- 3 i)$ .

Multiply $- 3$ by $- 4$ .

$\frac{2 - 6 i - 4 i + 12 i i}{(1 + 3 i) (1 - 3 i)}$

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

$\frac{2 - 6 i - 4 i + 12 (i^{1} i)}{(1 + 3 i) (1 - 3 i)}$

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

$\frac{2 - 6 i - 4 i + 12 (i^{1} i^{1})}{(1 + 3 i) (1 - 3 i)}$

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

$\frac{2 - 6 i - 4 i + 12 i^{1 + 1}}{(1 + 3 i) (1 - 3 i)}$

Add $1$ and $1$ .

$\frac{2 - 6 i - 4 i + 12 i^{2}}{(1 + 3 i) (1 - 3 i)}$

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

$\frac{2 - 6 i - 4 i + 12 \cdot - 1}{(1 + 3 i) (1 - 3 i)}$

Multiply $12$ by $- 1$ .

$\frac{2 - 6 i - 4 i - 12}{(1 + 3 i) (1 - 3 i)}$

Subtract $12$ from $2$ .

$\frac{- 10 - 6 i - 4 i}{(1 + 3 i) (1 - 3 i)}$

Subtract $4 i$ from $- 6 i$ .

$\frac{- 10 - 10 i}{(1 + 3 i) (1 - 3 i)}$

Simplify the denominator.

Expand $(1 + 3 i) (1 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 10 - 10 i}{1 (1 - 3 i) + 3 i (1 - 3 i)}$

Apply the distributive property.

$\frac{- 10 - 10 i}{1 \cdot 1 + 1 (- 3 i) + 3 i (1 - 3 i)}$

Apply the distributive property.

$\frac{- 10 - 10 i}{1 \cdot 1 + 1 (- 3 i) + 3 i \cdot 1 + 3 i (- 3 i)}$

Simplify.

Multiply $1$ by $1$ .

$\frac{- 10 - 10 i}{1 + 1 (- 3 i) + 3 i \cdot 1 + 3 i (- 3 i)}$

Multiply $- 3$ by $1$ .

$\frac{- 10 - 10 i}{1 - 3 i + 3 i \cdot 1 + 3 i (- 3 i)}$

Multiply $3$ by $1$ .

$\frac{- 10 - 10 i}{1 - 3 i + 3 i + 3 i (- 3 i)}$

Multiply $- 3$ by $3$ .

$\frac{- 10 - 10 i}{1 - 3 i + 3 i - 9 i i}$

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

$\frac{- 10 - 10 i}{1 - 3 i + 3 i - 9 (i^{1} i)}$

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

$\frac{- 10 - 10 i}{1 - 3 i + 3 i - 9 (i^{1} i^{1})}$

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

$\frac{- 10 - 10 i}{1 - 3 i + 3 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 10 - 10 i}{1 - 3 i + 3 i - 9 i^{2}}$

Add $- 3 i$ and $3 i$ .

$\frac{- 10 - 10 i}{1 + 0 - 9 i^{2}}$

Add $1$ and $0$ .

$\frac{- 10 - 10 i}{1 - 9 i^{2}}$

Simplify each term.

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

$\frac{- 10 - 10 i}{1 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{- 10 - 10 i}{1 + 9}$

Add $1$ and $9$ .

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

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

Factor $10$ out of $- 10$ .

$\frac{10 \cdot - 1 - 10 i}{10}$

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

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

Factor $10$ out of $10 (- 1) + 10 (- i)$ .

$\frac{10 (- 1 - i)}{10}$

Cancel the common factors.

Factor $10$ out of $10$ .

$\frac{10 (- 1 - i)}{10 (1)}$

Cancel the common factor.

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

Rewrite the expression.

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

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

$- 1 - i$

Simplify (2-4i)/(1+3i)

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