Precalculus Examples

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

Step 1

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $i i$ .

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Step 2.2.2.1

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

$\frac{2 i + i^{1} i}{3 i i}$

Step 2.2.2.2

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

$\frac{2 i + i^{1} i^{1}}{3 i i}$

Step 2.2.2.3

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

$\frac{2 i + i^{1 + 1}}{3 i i}$

Step 2.2.2.4

Add $1$ and $1$ .

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

Step 2.2.3

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

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

Step 2.2.4

Reorder $2 i$ and $- 1$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Add parentheses.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.3.6

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

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

Step 3

Multiply $3$ by $- 1$ .

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

Step 4

Split the fraction $\frac{- 1 + 2 i}{- 3}$ into two fractions.

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

Step 5

Simplify each term.

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Step 5.1

Dividing two negative values results in a positive value.

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

Step 5.2

Move the negative in front of the fraction.

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