Simplify 5/(2+4i)

$\frac{5}{2 + 4 i}$

Step 1

Multiply the numerator and denominator of $\frac{5}{2 + 4 i}$ by the conjugate of $2 + 4 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $5$ by $2$ .

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

Multiply $- 4$ by $5$ .

$\frac{10 - 20 i}{(2 + 4 i) (2 - 4 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{10 - 20 i}{2 (2 - 4 i) + 4 i (2 - 4 i)}$

Apply the distributive property.

$\frac{10 - 20 i}{2 \cdot 2 + 2 (- 4 i) + 4 i (2 - 4 i)}$

Apply the distributive property.

$\frac{10 - 20 i}{2 \cdot 2 + 2 (- 4 i) + 4 i \cdot 2 + 4 i (- 4 i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{10 - 20 i}{4 + 2 (- 4 i) + 4 i \cdot 2 + 4 i (- 4 i)}$

Multiply $- 4$ by $2$ .

$\frac{10 - 20 i}{4 - 8 i + 4 i \cdot 2 + 4 i (- 4 i)}$

Multiply $2$ by $4$ .

$\frac{10 - 20 i}{4 - 8 i + 8 i + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

$\frac{10 - 20 i}{4 - 8 i + 8 i - 16 i i}$

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

$\frac{10 - 20 i}{4 - 8 i + 8 i - 16 (i^{1} i)}$

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

$\frac{10 - 20 i}{4 - 8 i + 8 i - 16 (i^{1} i^{1})}$

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

$\frac{10 - 20 i}{4 - 8 i + 8 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{10 - 20 i}{4 - 8 i + 8 i - 16 i^{2}}$

Add $- 8 i$ and $8 i$ .

$\frac{10 - 20 i}{4 + 0 - 16 i^{2}}$

Add $4$ and $0$ .

$\frac{10 - 20 i}{4 - 16 i^{2}}$

Simplify each term.

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

$\frac{10 - 20 i}{4 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{10 - 20 i}{4 + 16}$

Add $4$ and $16$ .

$\frac{10 - 20 i}{20}$

Step 3

Cancel the common factor of $10 - 20 i$ and $20$ .

Factor $10$ out of $10$ .

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

Factor $10$ out of $- 20 i$ .

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

Factor $10$ out of $10 (1) + 10 (- 2 i)$ .

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

Cancel the common factors.

Factor $10$ out of $20$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Split the fraction $\frac{1 - 2 i}{2}$ into two fractions.

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

Step 5

Cancel the common factor of $- 2$ and $2$ .

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- i$ by $1$ .

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

Simplify 5/(2+4i)

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