Simplify (2i)/(3+i)

$\frac{2 i}{3 + i}$

Step 1

Multiply the numerator and denominator of $\frac{2 i}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

$\frac{2 i}{3 + 1 i} \cdot \frac{3 - i}{3 - i}$

Step 2

Multiply.

Combine.

$\frac{2 i (3 - i)}{(3 + 1 i) (3 - i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{2 i \cdot 3 + 2 i (- i)}{(3 + 1 i) (3 - i)}$

Multiply $3$ by $2$ .

$\frac{6 i + 2 i (- i)}{(3 + 1 i) (3 - i)}$

Multiply $2 i (- i)$ .

Multiply $- 1$ by $2$ .

$\frac{6 i - 2 i i}{(3 + 1 i) (3 - i)}$

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

$\frac{6 i - 2 (i^{1} i)}{(3 + 1 i) (3 - i)}$

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

$\frac{6 i - 2 (i^{1} i^{1})}{(3 + 1 i) (3 - i)}$

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

$\frac{6 i - 2 i^{1 + 1}}{(3 + 1 i) (3 - i)}$

Add $1$ and $1$ .

$\frac{6 i - 2 i^{2}}{(3 + 1 i) (3 - i)}$

Simplify each term.

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

$\frac{6 i - 2 \cdot - 1}{(3 + 1 i) (3 - i)}$

Multiply $- 2$ by $- 1$ .

$\frac{6 i + 2}{(3 + 1 i) (3 - i)}$

Reorder $6 i$ and $2$ .

$\frac{2 + 6 i}{(3 + 1 i) (3 - i)}$

Simplify the denominator.

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 + 6 i}{3 (3 - i) + 1 i (3 - i)}$

Apply the distributive property.

$\frac{2 + 6 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}$

Apply the distributive property.

$\frac{2 + 6 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{2 + 6 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Multiply $- 1$ by $3$ .

$\frac{2 + 6 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}$

Multiply $3$ by $1$ .

$\frac{2 + 6 i}{9 - 3 i + 3 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{2 + 6 i}{9 - 3 i + 3 i - i i}$

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

$\frac{2 + 6 i}{9 - 3 i + 3 i - (i^{1} i)}$

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

$\frac{2 + 6 i}{9 - 3 i + 3 i - (i^{1} i^{1})}$

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

$\frac{2 + 6 i}{9 - 3 i + 3 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{2 + 6 i}{9 - 3 i + 3 i - i^{2}}$

Add $- 3 i$ and $3 i$ .

$\frac{2 + 6 i}{9 + 0 - i^{2}}$

Add $9$ and $0$ .

$\frac{2 + 6 i}{9 - i^{2}}$

Simplify each term.

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

$\frac{2 + 6 i}{9 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{2 + 6 i}{9 + 1}$

Add $9$ and $1$ .

$\frac{2 + 6 i}{10}$

Step 3

Cancel the common factor of $2 + 6 i$ and $10$ .

Factor $2$ out of $2$ .

$\frac{2 (1) + 6 i}{10}$

Factor $2$ out of $6 i$ .

$\frac{2 (1) + 2 (3 i)}{10}$

Factor $2$ out of $2 (1) + 2 (3 i)$ .

$\frac{2 (1 + 3 i)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 (1 + 3 i)}{2 \cdot 5}$

Cancel the common factor.

$\frac{2 (1 + 3 i)}{2 \cdot 5}$

Rewrite the expression.

$\frac{1 + 3 i}{5}$

Step 4

Split the fraction $\frac{1 + 3 i}{5}$ into two fractions.

$\frac{1}{5} + \frac{3 i}{5}$

Simplify (2i)/(3+i)

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