Simplify 4/(5+i)

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

Step 1

Multiply the numerator and denominator of $\frac{4}{5 + 1 i}$ by the conjugate of $5 + 1 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $4$ by $5$ .

$\frac{20 + 4 (- i)}{(5 + 1 i) (5 - i)}$

Multiply $- 1$ by $4$ .

$\frac{20 - 4 i}{(5 + 1 i) (5 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{20 - 4 i}{5 (5 - i) + 1 i (5 - i)}$

Apply the distributive property.

$\frac{20 - 4 i}{5 \cdot 5 + 5 (- i) + 1 i (5 - i)}$

Apply the distributive property.

$\frac{20 - 4 i}{5 \cdot 5 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{20 - 4 i}{25 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}$

Multiply $- 1$ by $5$ .

$\frac{20 - 4 i}{25 - 5 i + 1 i \cdot 5 + 1 i (- i)}$

Multiply $5$ by $1$ .

$\frac{20 - 4 i}{25 - 5 i + 5 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{20 - 4 i}{25 - 5 i + 5 i - i i}$

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

$\frac{20 - 4 i}{25 - 5 i + 5 i - (i^{1} i)}$

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

$\frac{20 - 4 i}{25 - 5 i + 5 i - (i^{1} i^{1})}$

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

$\frac{20 - 4 i}{25 - 5 i + 5 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{20 - 4 i}{25 - 5 i + 5 i - i^{2}}$

Add $- 5 i$ and $5 i$ .

$\frac{20 - 4 i}{25 + 0 - i^{2}}$

Add $25$ and $0$ .

$\frac{20 - 4 i}{25 - i^{2}}$

Simplify each term.

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

$\frac{20 - 4 i}{25 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{20 - 4 i}{25 + 1}$

Add $25$ and $1$ .

$\frac{20 - 4 i}{26}$

Step 3

Cancel the common factor of $20 - 4 i$ and $26$ .

Factor $2$ out of $20$ .

$\frac{2 (10) - 4 i}{26}$

Factor $2$ out of $- 4 i$ .

$\frac{2 (10) + 2 (- 2 i)}{26}$

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

$\frac{2 (10 - 2 i)}{26}$

Cancel the common factors.

Factor $2$ out of $26$ .

$\frac{2 (10 - 2 i)}{2 \cdot 13}$

Cancel the common factor.

$\frac{2 (10 - 2 i)}{2 \cdot 13}$

Rewrite the expression.

$\frac{10 - 2 i}{13}$

Step 4

Split the fraction $\frac{10 - 2 i}{13}$ into two fractions.

$\frac{10}{13} + \frac{- 2 i}{13}$

Step 5

Move the negative in front of the fraction.

$\frac{10}{13} - \frac{2 i}{13}$

Simplify 4/(5+i)

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