Precalculus Examples

$\frac{- 3 + 5 i}{15 i}$

Step 1

Multiply the numerator and denominator of $\frac{- 3 + 5 i}{15 i}$ by the conjugate of $15 i$ to make the denominator real.

$\frac{- 3 + 5 i}{15 i} \cdot \frac{i}{i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(- 3 + 5 i) i}{15 i i}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{- 3 i + 5 i i}{15 i i}$

Step 2.2.2

Multiply $5 i i$ .

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Step 2.2.2.1

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

$\frac{- 3 i + 5 (i^{1} i)}{15 i i}$

Step 2.2.2.2

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

$\frac{- 3 i + 5 (i^{1} i^{1})}{15 i i}$

Step 2.2.2.3

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

$\frac{- 3 i + 5 i^{1 + 1}}{15 i i}$

Step 2.2.2.4

Add $1$ and $1$ .

$\frac{- 3 i + 5 i^{2}}{15 i i}$

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

$\frac{- 3 i + 5 \cdot - 1}{15 i i}$

Step 2.2.3.2

Multiply $5$ by $- 1$ .

$\frac{- 3 i - 5}{15 i i}$

Step 2.2.4

Reorder $- 3 i$ and $- 5$ .

$\frac{- 5 - 3 i}{15 i i}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Add parentheses.

$\frac{- 5 - 3 i}{15 (i i)}$

Step 2.3.2

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

$\frac{- 5 - 3 i}{15 (i^{1} i)}$

Step 2.3.3

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

$\frac{- 5 - 3 i}{15 (i^{1} i^{1})}$

Step 2.3.4

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

$\frac{- 5 - 3 i}{15 i^{1 + 1}}$

Step 2.3.5

Add $1$ and $1$ .

$\frac{- 5 - 3 i}{15 i^{2}}$

Step 2.3.6

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

$\frac{- 5 - 3 i}{15 \cdot - 1}$

Step 3

Multiply $15$ by $- 1$ .

$\frac{- 5 - 3 i}{- 15}$

Step 4

Split the fraction $\frac{- 5 - 3 i}{- 15}$ into two fractions.

$\frac{- 5}{- 15} + \frac{- 3 i}{- 15}$

Step 5

Simplify each term.

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Step 5.1

Cancel the common factor of $- 5$ and $- 15$ .

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Step 5.1.1

Factor $- 5$ out of $- 5$ .

$\frac{- 5 (1)}{- 15} + \frac{- 3 i}{- 15}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $- 5$ out of $- 15$ .

$\frac{- 5 \cdot 1}{- 5 \cdot 3} + \frac{- 3 i}{- 15}$

Step 5.1.2.2

Cancel the common factor.

$\frac{- 5 \cdot 1}{- 5 \cdot 3} + \frac{- 3 i}{- 15}$

Step 5.1.2.3

Rewrite the expression.

$\frac{1}{3} + \frac{- 3 i}{- 15}$

Step 5.2

Cancel the common factor of $- 3$ and $- 15$ .

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Step 5.2.1

Factor $- 3$ out of $- 3 i$ .

$\frac{1}{3} + \frac{- 3 (i)}{- 15}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $- 3$ out of $- 15$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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