Precalculus Examples

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

Step 1

Expand $(\frac{1}{2} - i) (\frac{1}{4} + 2 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\frac{1}{2} \cdot \frac{1}{4}$ .

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Step 2.1.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

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

Step 2.1.1.2

Multiply $2$ by $4$ .

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

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

Factor $2$ out of $2 i$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.1.3

Combine $\frac{1}{4}$ and $i$ .

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

Step 2.1.4

Multiply $- i (2 i)$ .

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Step 2.1.4.1

Multiply $2$ by $- 1$ .

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

Step 2.1.4.2

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

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

Step 2.1.4.3

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

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

Step 2.1.4.4

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

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

Step 2.1.4.5

Add $1$ and $1$ .

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

Step 2.1.5

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

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

Step 2.1.6

Multiply $- 2$ by $- 1$ .

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

Step 2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

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

Step 2.3

Combine $2$ and $\frac{8}{8}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $2$ by $8$ .

$\frac{1 + 16}{8} + i - \frac{i}{4}$

Step 2.5.2

Add $1$ and $16$ .

$\frac{17}{8} + i - \frac{i}{4}$

Step 2.6

To write $i$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{17}{8} + i \cdot \frac{4}{4} - \frac{i}{4}$

Step 2.7

Combine $i$ and $\frac{4}{4}$ .

$\frac{17}{8} + \frac{i \cdot 4}{4} - \frac{i}{4}$

Step 2.8

Combine the numerators over the common denominator.

$\frac{17}{8} + \frac{3 i}{4}$