Algebra Examples

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

Step 1

Multiply the numerator and denominator of $\frac{- 9 - 2 i}{- 3 i}$ by the conjugate of $- 3 i$ to make the denominator real.

$\frac{- 9 - 2 i}{- 3 i} \cdot \frac{i}{i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $- 2 i i$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Add $1$ and $1$ .

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

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

$\frac{- 9 i - 2 \cdot - 1}{- 3 i i}$

Step 2.2.3.2

Multiply $- 2$ by $- 1$ .

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

Step 2.2.4

Reorder $- 9 i$ and $2$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Add parentheses.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.3.6

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

$\frac{2 - 9 i}{- 3 \cdot - 1}$

Step 3

Multiply $- 3$ by $- 1$ .

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

Step 4

Split the fraction $\frac{2 - 9 i}{3}$ into two fractions.

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

Step 5

Cancel the common factor of $- 9$ and $3$ .

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Step 5.1

Factor $3$ out of $- 9 i$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $3$ out of $3$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.2.4

Divide $- 3 i$ by $1$ .

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