Evaluate 4/(4+i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $4$ by $4$ .

$\frac{16 + 4 (- i)}{(4 + 1 i) (4 - i)}$

Multiply $- 1$ by $4$ .

$\frac{16 - 4 i}{(4 + 1 i) (4 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{16 - 4 i}{4 (4 - i) + 1 i (4 - i)}$

Apply the distributive property.

$\frac{16 - 4 i}{4 \cdot 4 + 4 (- i) + 1 i (4 - i)}$

Apply the distributive property.

$\frac{16 - 4 i}{4 \cdot 4 + 4 (- i) + 1 i \cdot 4 + 1 i (- i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{16 - 4 i}{16 + 4 (- i) + 1 i \cdot 4 + 1 i (- i)}$

Multiply $- 1$ by $4$ .

$\frac{16 - 4 i}{16 - 4 i + 1 i \cdot 4 + 1 i (- i)}$

Multiply $4$ by $1$ .

$\frac{16 - 4 i}{16 - 4 i + 4 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Add $- 4 i$ and $4 i$ .

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

Add $16$ and $0$ .

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

Simplify each term.

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

$\frac{16 - 4 i}{16 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{16 - 4 i}{16 + 1}$

Add $16$ and $1$ .

$\frac{16 - 4 i}{17}$

Step 3

Split the fraction $\frac{16 - 4 i}{17}$ into two fractions.

$\frac{16}{17} + \frac{- 4 i}{17}$

Step 4

Move the negative in front of the fraction.

$\frac{16}{17} - \frac{4 i}{17}$

Evaluate 4/(4+i)

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