Trigonometry Examples

$\frac{18 + 9 i}{- 4 i}$

Step 1

Multiply the numerator and denominator of $\frac{18 + 9 i}{- 4 i}$ by the conjugate of $- 4 i$ to make the denominator real.

$\frac{18 + 9 i}{- 4 i} \cdot \frac{i}{i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(18 + 9 i) i}{- 4 i i}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{18 i + 9 i i}{- 4 i i}$

Step 2.2.2

Multiply $9 i i$ .

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Step 2.2.2.1

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

$\frac{18 i + 9 (i^{1} i)}{- 4 i i}$

Step 2.2.2.2

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

$\frac{18 i + 9 (i^{1} i^{1})}{- 4 i i}$

Step 2.2.2.3

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

$\frac{18 i + 9 i^{1 + 1}}{- 4 i i}$

Step 2.2.2.4

Add $1$ and $1$ .

$\frac{18 i + 9 i^{2}}{- 4 i i}$

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

$\frac{18 i + 9 \cdot - 1}{- 4 i i}$

Step 2.2.3.2

Multiply $9$ by $- 1$ .

$\frac{18 i - 9}{- 4 i i}$

Step 2.2.4

Reorder $18 i$ and $- 9$ .

$\frac{- 9 + 18 i}{- 4 i i}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Add parentheses.

$\frac{- 9 + 18 i}{- 4 (i i)}$

Step 2.3.2

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

$\frac{- 9 + 18 i}{- 4 (i^{1} i)}$

Step 2.3.3

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

$\frac{- 9 + 18 i}{- 4 (i^{1} i^{1})}$

Step 2.3.4

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

$\frac{- 9 + 18 i}{- 4 i^{1 + 1}}$

Step 2.3.5

Add $1$ and $1$ .

$\frac{- 9 + 18 i}{- 4 i^{2}}$

Step 2.3.6

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

$\frac{- 9 + 18 i}{- 4 \cdot - 1}$

Step 3

Multiply $- 4$ by $- 1$ .

$\frac{- 9 + 18 i}{4}$

Step 4

Split the fraction $\frac{- 9 + 18 i}{4}$ into two fractions.

$\frac{- 9}{4} + \frac{18 i}{4}$

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

$- \frac{9}{4} + \frac{18 i}{4}$

Step 5.2

Cancel the common factor of $18$ and $4$ .

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Step 5.2.1

Factor $2$ out of $18 i$ .

$- \frac{9}{4} + \frac{2 (9 i)}{4}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $4$ .

$- \frac{9}{4} + \frac{2 (9 i)}{2 (2)}$

Step 5.2.2.2

Cancel the common factor.

$- \frac{9}{4} + \frac{2 (9 i)}{2 \cdot 2}$

Step 5.2.2.3

Rewrite the expression.

$- \frac{9}{4} + \frac{9 i}{2}$