Precalculus Examples

$\frac{8 i}{6 - 9 i}$

Step 1

Multiply the numerator and denominator of $\frac{8 i}{6 - 9 i}$ by the conjugate of $6 - 9 i$ to make the denominator real.

$\frac{8 i}{6 - 9 i} \cdot \frac{6 + 9 i}{6 + 9 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{8 i (6 + 9 i)}{(6 - 9 i) (6 + 9 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{8 i \cdot 6 + 8 i (9 i)}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.2

Multiply $6$ by $8$ .

$\frac{48 i + 8 i (9 i)}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.3

Multiply $8 i (9 i)$ .

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Step 2.2.3.1

Multiply $9$ by $8$ .

$\frac{48 i + 72 i i}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.3.2

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

$\frac{48 i + 72 (i^{1} i)}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.3.3

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

$\frac{48 i + 72 (i^{1} i^{1})}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.3.4

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

$\frac{48 i + 72 i^{1 + 1}}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.3.5

Add $1$ and $1$ .

$\frac{48 i + 72 i^{2}}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

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

$\frac{48 i + 72 \cdot - 1}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.4.2

Multiply $72$ by $- 1$ .

$\frac{48 i - 72}{(6 - 9 i) (6 + 9 i)}$

Step 2.2.5

Reorder $48 i$ and $- 72$ .

$\frac{- 72 + 48 i}{(6 - 9 i) (6 + 9 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(6 - 9 i) (6 + 9 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 72 + 48 i}{6 (6 + 9 i) - 9 i (6 + 9 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 72 + 48 i}{6 \cdot 6 + 6 (9 i) - 9 i (6 + 9 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 72 + 48 i}{6 \cdot 6 + 6 (9 i) - 9 i \cdot 6 - 9 i (9 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $6$ by $6$ .

$\frac{- 72 + 48 i}{36 + 6 (9 i) - 9 i \cdot 6 - 9 i (9 i)}$

Step 2.3.2.2

Multiply $9$ by $6$ .

$\frac{- 72 + 48 i}{36 + 54 i - 9 i \cdot 6 - 9 i (9 i)}$

Step 2.3.2.3

Multiply $6$ by $- 9$ .

$\frac{- 72 + 48 i}{36 + 54 i - 54 i - 9 i (9 i)}$

Step 2.3.2.4

Multiply $9$ by $- 9$ .

$\frac{- 72 + 48 i}{36 + 54 i - 54 i - 81 i i}$

Step 2.3.2.5

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

$\frac{- 72 + 48 i}{36 + 54 i - 54 i - 81 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 72 + 48 i}{36 + 54 i - 54 i - 81 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 72 + 48 i}{36 + 54 i - 54 i - 81 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 72 + 48 i}{36 + 54 i - 54 i - 81 i^{2}}$

Step 2.3.2.9

Subtract $54 i$ from $54 i$ .

$\frac{- 72 + 48 i}{36 + 0 - 81 i^{2}}$

Step 2.3.2.10

Add $36$ and $0$ .

$\frac{- 72 + 48 i}{36 - 81 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 72 + 48 i}{36 - 81 \cdot - 1}$

Step 2.3.3.2

Multiply $- 81$ by $- 1$ .

$\frac{- 72 + 48 i}{36 + 81}$

Step 2.3.4

Add $36$ and $81$ .

$\frac{- 72 + 48 i}{117}$

Step 3

Cancel the common factor of $- 72 + 48 i$ and $117$ .

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Step 3.1

Factor $3$ out of $- 72$ .

$\frac{3 (- 24) + 48 i}{117}$

Step 3.2

Factor $3$ out of $48 i$ .

$\frac{3 (- 24) + 3 (16 i)}{117}$

Step 3.3

Factor $3$ out of $3 (- 24) + 3 (16 i)$ .

$\frac{3 (- 24 + 16 i)}{117}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $3$ out of $117$ .

$\frac{3 (- 24 + 16 i)}{3 \cdot 39}$

Step 3.4.2

Cancel the common factor.

$\frac{3 (- 24 + 16 i)}{3 \cdot 39}$

Step 3.4.3

Rewrite the expression.

$\frac{- 24 + 16 i}{39}$

Step 4

Split the fraction $\frac{- 24 + 16 i}{39}$ into two fractions.

$\frac{- 24}{39} + \frac{16 i}{39}$

Step 5

Simplify each term.

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Step 5.1

Cancel the common factor of $- 24$ and $39$ .

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Step 5.1.1

Factor $3$ out of $- 24$ .

$\frac{3 (- 8)}{39} + \frac{16 i}{39}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $3$ out of $39$ .

$\frac{3 \cdot - 8}{3 \cdot 13} + \frac{16 i}{39}$

Step 5.1.2.2

Cancel the common factor.

$\frac{3 \cdot - 8}{3 \cdot 13} + \frac{16 i}{39}$

Step 5.1.2.3

Rewrite the expression.

$\frac{- 8}{13} + \frac{16 i}{39}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{8}{13} + \frac{16 i}{39}$