Precalculus Examples

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

Step 1

Multiply the numerator and denominator of $\frac{4 - i}{2 - i}$ by the conjugate of $2 - i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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 $4$ by $2$ .

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

Step 2.2.2.1.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2.1.3

Multiply $- i i$ .

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Step 2.2.2.1.3.1

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

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

Step 2.2.2.1.3.2

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

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

Step 2.2.2.1.3.3

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

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

Step 2.2.2.1.3.4

Add $1$ and $1$ .

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

Step 2.2.2.1.4

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

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

Step 2.2.2.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2.2

Add $8$ and $1$ .

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

Step 2.2.2.3

Subtract $2 i$ from $4 i$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $2$ by $2$ .

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

Step 2.3.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.2.3

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

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

Step 2.3.2.4

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

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

Step 2.3.2.5

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

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

Step 2.3.2.6

Add $1$ and $1$ .

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

Step 2.3.2.7

Subtract $2 i$ from $2 i$ .

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

Step 2.3.2.8

Add $4$ and $0$ .

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4

Add $4$ and $1$ .

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

Step 3

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

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