Simplify i/(1+2i)

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

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

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

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $i$ by $1$ .

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

Multiply $i (- 2 i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

Multiply $- 2$ by $- 1$ .

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

Reorder $i$ and $2$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $1$ by $1$ .

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

Multiply $- 2$ by $1$ .

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

Multiply $2$ by $1$ .

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

Multiply $- 2$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Add $1$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 4$ by $- 1$ .

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

Add $1$ and $4$ .

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

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

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

Simplify i/(1+2i)

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