Simplify i/(2+i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Move $2$ to the left of $i$ .

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

Multiply $i (- i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Reorder $2 i$ and $1$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Add $4$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $4$ and $1$ .

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

Step 3

Split the fraction $\frac{1 + 2 i}{5}$ into two fractions.

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

Simplify i/(2+i)

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