Precalculus Examples

\frac{4 - i}{2 - i}

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

Step 1

Multiply the numerator and denominator of

\frac{4 - i}{2 - i}

$\frac{4 - i}{2 - i}$ by the conjugate of

2 - i

$2 - i$ to make the denominator real.

\frac{4 - i}{2 - i} \cdot \frac{2 + i}{2 + i}

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

Step 2

Multiply.

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Step 2.1

Combine.

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(4 - i) (2 + i)

$(4 - i) (2 + i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

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

Step 2.2.1.2

Apply the distributive property.

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

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

Step 2.2.1.3

Apply the distributive property.

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

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

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

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

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

4

$4$ by

2

$2$ .

\frac{8 + 4 i - i \cdot 2 - i i}{(2 - i) (2 + i)}

$\frac{8 + 4 i - i \cdot 2 - i i}{(2 - i) (2 + i)}$

Step 2.2.2.1.2

Multiply

2

$2$ by

- 1

$- 1$ .

\frac{8 + 4 i - 2 i - i i}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i - i i}{(2 - i) (2 + i)}$

Step 2.2.2.1.3

Multiply

- i i

$- i i$ .

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Step 2.2.2.1.3.1

Raise

i

$i$ to the power of

1

$1$ .

\frac{8 + 4 i - 2 i - (i^{1} i)}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i - (i^{1} i)}{(2 - i) (2 + i)}$

Step 2.2.2.1.3.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{8 + 4 i - 2 i - (i^{1} i^{1})}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i - (i^{1} i^{1})}{(2 - i) (2 + i)}$

Step 2.2.2.1.3.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{8 + 4 i - 2 i - i^{1 + 1}}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i - i^{1 + 1}}{(2 - i) (2 + i)}$

Step 2.2.2.1.3.4

Add

1

$1$ and

1

$1$ .

\frac{8 + 4 i - 2 i - i^{2}}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i - i^{2}}{(2 - i) (2 + i)}$

\frac{8 + 4 i - 2 i - i^{2}}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i - i^{2}}{(2 - i) (2 + i)}$

Step 2.2.2.1.4

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{8 + 4 i - 2 i - - 1}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i - - 1}{(2 - i) (2 + i)}$

Step 2.2.2.1.5

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{8 + 4 i - 2 i + 1}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i + 1}{(2 - i) (2 + i)}$

\frac{8 + 4 i - 2 i + 1}{(2 - i) (2 + i)}

$\frac{8 + 4 i - 2 i + 1}{(2 - i) (2 + i)}$

Step 2.2.2.2

Add

8

$8$ and

1

$1$ .

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

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

Step 2.2.2.3

Subtract

2 i

$2 i$ from

4 i

$4 i$ .

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

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

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

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

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

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(2 - i) (2 + i)

$(2 - i) (2 + i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

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

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

Step 2.3.1.2

Apply the distributive property.

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

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

Step 2.3.1.3

Apply the distributive property.

\frac{9 + 2 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}

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

\frac{9 + 2 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

2

$2$ by

2

$2$ .

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

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

Step 2.3.2.2

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 2.3.2.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{9 + 2 i}{4 + 2 i - 2 i - (i^{1} i)}

$\frac{9 + 2 i}{4 + 2 i - 2 i - (i^{1} i)}$

Step 2.3.2.4

Raise

i

$i$ to the power of

1

$1$ .

\frac{9 + 2 i}{4 + 2 i - 2 i - (i^{1} i^{1})}

$\frac{9 + 2 i}{4 + 2 i - 2 i - (i^{1} i^{1})}$

Step 2.3.2.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{9 + 2 i}{4 + 2 i - 2 i - i^{1 + 1}}

$\frac{9 + 2 i}{4 + 2 i - 2 i - i^{1 + 1}}$

Step 2.3.2.6

Add

1

$1$ and

1

$1$ .

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

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

Step 2.3.2.7

Subtract

2 i

$2 i$ from

2 i

$2 i$ .

\frac{9 + 2 i}{4 + 0 - i^{2}}

$\frac{9 + 2 i}{4 + 0 - i^{2}}$

Step 2.3.2.8

Add

4

$4$ and

0

$0$ .

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

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

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

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{9 + 2 i}{4 - - 1}

$\frac{9 + 2 i}{4 - - 1}$

Step 2.3.3.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{9 + 2 i}{4 + 1}

$\frac{9 + 2 i}{4 + 1}$

\frac{9 + 2 i}{4 + 1}

$\frac{9 + 2 i}{4 + 1}$

Step 2.3.4

Add

4

$4$ and

1

$1$ .

\frac{9 + 2 i}{5}

$\frac{9 + 2 i}{5}$

\frac{9 + 2 i}{5}

$\frac{9 + 2 i}{5}$

\frac{9 + 2 i}{5}

$\frac{9 + 2 i}{5}$

Step 3

Split the fraction

\frac{9 + 2 i}{5}

$\frac{9 + 2 i}{5}$ into two fractions.

\frac{9}{5} + \frac{2 i}{5}

$\frac{9}{5} + \frac{2 i}{5}$