Precalculus Examples

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

Step 1

Multiply the numerator and denominator of $\frac{8 - i}{9 + 1 i}$ by the conjugate of $9 + 1 i$ to make the denominator real.

$\frac{8 - i}{9 + 1 i} \cdot \frac{9 - i}{9 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(8 - i) (9 - i)}{(9 + 1 i) (9 - i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(8 - i) (9 - i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{8 (9 - i) - i (9 - i)}{(9 + 1 i) (9 - i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{8 \cdot 9 + 8 (- i) - i (9 - i)}{(9 + 1 i) (9 - i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{8 \cdot 9 + 8 (- i) - i \cdot 9 - i (- i)}{(9 + 1 i) (9 - i)}$

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $8$ by $9$ .

$\frac{72 + 8 (- i) - i \cdot 9 - i (- i)}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.2

Multiply $- 1$ by $8$ .

$\frac{72 - 8 i - i \cdot 9 - i (- i)}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.3

Multiply $9$ by $- 1$ .

$\frac{72 - 8 i - 9 i - i (- i)}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.4

Multiply $- i (- i)$ .

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Step 2.2.2.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{72 - 8 i - 9 i + 1 i i}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.4.2

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

$\frac{72 - 8 i - 9 i + 1 (i^{1} i)}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.4.3

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

$\frac{72 - 8 i - 9 i + 1 (i^{1} i^{1})}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.4.4

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

$\frac{72 - 8 i - 9 i + 1 i^{1 + 1}}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{72 - 8 i - 9 i + 1 i^{2}}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.4.6

Multiply $i^{2}$ by $1$ .

$\frac{72 - 8 i - 9 i + i^{2}}{(9 + 1 i) (9 - i)}$

Step 2.2.2.1.5

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

$\frac{72 - 8 i - 9 i - 1}{(9 + 1 i) (9 - i)}$

Step 2.2.2.2

Subtract $1$ from $72$ .

$\frac{71 - 8 i - 9 i}{(9 + 1 i) (9 - i)}$

Step 2.2.2.3

Subtract $9 i$ from $- 8 i$ .

$\frac{71 - 17 i}{(9 + 1 i) (9 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(9 + 1 i) (9 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{71 - 17 i}{9 (9 - i) + 1 i (9 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{71 - 17 i}{9 \cdot 9 + 9 (- i) + 1 i (9 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{71 - 17 i}{9 \cdot 9 + 9 (- i) + 1 i \cdot 9 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $9$ by $9$ .

$\frac{71 - 17 i}{81 + 9 (- i) + 1 i \cdot 9 + 1 i (- i)}$

Step 2.3.2.2

Multiply $- 1$ by $9$ .

$\frac{71 - 17 i}{81 - 9 i + 1 i \cdot 9 + 1 i (- i)}$

Step 2.3.2.3

Multiply $9$ by $1$ .

$\frac{71 - 17 i}{81 - 9 i + 9 i + 1 i (- i)}$

Step 2.3.2.4

Multiply $- 1$ by $1$ .

$\frac{71 - 17 i}{81 - 9 i + 9 i - i i}$

Step 2.3.2.5

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

$\frac{71 - 17 i}{81 - 9 i + 9 i - (i^{1} i)}$

Step 2.3.2.6

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

$\frac{71 - 17 i}{81 - 9 i + 9 i - (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{71 - 17 i}{81 - 9 i + 9 i - i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{71 - 17 i}{81 - 9 i + 9 i - i^{2}}$

Step 2.3.2.9

Add $- 9 i$ and $9 i$ .

$\frac{71 - 17 i}{81 + 0 - i^{2}}$

Step 2.3.2.10

Add $81$ and $0$ .

$\frac{71 - 17 i}{81 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{71 - 17 i}{81 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{71 - 17 i}{81 + 1}$

Step 2.3.4

Add $81$ and $1$ .

$\frac{71 - 17 i}{82}$

Step 3

Split the fraction $\frac{71 - 17 i}{82}$ into two fractions.

$\frac{71}{82} + \frac{- 17 i}{82}$

Step 4

Move the negative in front of the fraction.

$\frac{71}{82} - \frac{17 i}{82}$