Simplify (3i)/(-2+i)

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

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

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

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 2$ by $3$ .

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

Multiply $3 i (- i)$ .

Multiply $- 1$ by $3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

Multiply $- 3$ by $- 1$ .

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

Reorder $- 6 i$ and $3$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $- 2$ by $- 2$ .

$\frac{3 - 6 i}{4 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)}$

Multiply $- 1$ by $- 2$ .

$\frac{3 - 6 i}{4 + 2 i + 1 i \cdot - 2 + 1 i (- i)}$

Multiply $- 2$ by $1$ .

$\frac{3 - 6 i}{4 + 2 i - 2 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{3 - 6 i}{4 + 2 i - 2 i - i i}$

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

$\frac{3 - 6 i}{4 + 2 i - 2 i - (i^{1} i)}$

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

$\frac{3 - 6 i}{4 + 2 i - 2 i - (i^{1} i^{1})}$

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

$\frac{3 - 6 i}{4 + 2 i - 2 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3 - 6 i}{4 + 2 i - 2 i - i^{2}}$

Subtract $2 i$ from $2 i$ .

$\frac{3 - 6 i}{4 + 0 - i^{2}}$

Add $4$ and $0$ .

$\frac{3 - 6 i}{4 - i^{2}}$

Simplify each term.

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

$\frac{3 - 6 i}{4 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{3 - 6 i}{4 + 1}$

Add $4$ and $1$ .

$\frac{3 - 6 i}{5}$

Split the fraction $\frac{3 - 6 i}{5}$ into two fractions.

$\frac{3}{5} + \frac{- 6 i}{5}$

Move the negative in front of the fraction.

$\frac{3}{5} - \frac{6 i}{5}$

Simplify (3i)/(-2+i)

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