Basic Math Examples

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

Step 1

Expand $(\frac{1}{2} - 9 i) (\frac{1}{2} + 9 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

Multiply $2$ by $2$ .

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

Step 2.1.2

Multiply $\frac{1}{2} (9 i)$ .

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Step 2.1.2.1

Combine $9$ and $\frac{1}{2}$ .

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

Step 2.1.2.2

Combine $\frac{9}{2}$ and $i$ .

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

Step 2.1.3

Multiply $- 9 i \frac{1}{2}$ .

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Step 2.1.3.1

Combine $\frac{1}{2}$ and $- 9$ .

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

Step 2.1.3.2

Combine $\frac{- 9}{2}$ and $i$ .

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

Step 2.1.4

Move the negative in front of the fraction.

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

Step 2.1.5

Multiply $- 9 i (9 i)$ .

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Step 2.1.5.1

Multiply $9$ by $- 9$ .

$\frac{1}{4} + \frac{9 i}{2} - \frac{9 i}{2} - 81 i i$

Step 2.1.5.2

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

$\frac{1}{4} + \frac{9 i}{2} - \frac{9 i}{2} - 81 (i^{1} i)$

Step 2.1.5.3

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

$\frac{1}{4} + \frac{9 i}{2} - \frac{9 i}{2} - 81 (i^{1} i^{1})$

Step 2.1.5.4

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

$\frac{1}{4} + \frac{9 i}{2} - \frac{9 i}{2} - 81 i^{1 + 1}$

Step 2.1.5.5

Add $1$ and $1$ .

$\frac{1}{4} + \frac{9 i}{2} - \frac{9 i}{2} - 81 i^{2}$

Step 2.1.6

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

$\frac{1}{4} + \frac{9 i}{2} - \frac{9 i}{2} - 81 \cdot - 1$

Step 2.1.7

Multiply $- 81$ by $- 1$ .

$\frac{1}{4} + \frac{9 i}{2} - \frac{9 i}{2} + 81$

Step 2.2

To write $81$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{4} + 81 \cdot \frac{4}{4} + \frac{9 i}{2} - \frac{9 i}{2}$

Step 2.3

Combine $81$ and $\frac{4}{4}$ .

$\frac{1}{4} + \frac{81 \cdot 4}{4} + \frac{9 i}{2} - \frac{9 i}{2}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1 + 81 \cdot 4}{4} + \frac{9 i}{2} - \frac{9 i}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $81$ by $4$ .

$\frac{1 + 324}{4} + \frac{9 i}{2} - \frac{9 i}{2}$

Step 2.5.2

Add $1$ and $324$ .

$\frac{325}{4} + \frac{9 i}{2} - \frac{9 i}{2}$

Step 2.6

Combine the numerators over the common denominator.

$\frac{325}{4} + \frac{0}{2}$

Step 2.7

To write $\frac{0}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{325}{4} + \frac{0}{2} \cdot \frac{2}{2}$

Step 2.8

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.8.1

Multiply $\frac{0}{2}$ by $\frac{2}{2}$ .

$\frac{325}{4} + \frac{0 \cdot 2}{2 \cdot 2}$

Step 2.8.2

Multiply $2$ by $2$ .

$\frac{325}{4} + \frac{0 \cdot 2}{4}$

Step 2.9

Combine the numerators over the common denominator.

$\frac{325 + 0 \cdot 2}{4}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $0$ by $2$ .

$\frac{325 + 0}{4}$

Step 2.10.2

Add $325$ and $0$ .

$\frac{325}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{325}{4}$

Decimal Form:

$81.25$

Mixed Number Form:

$81 \frac{1}{4}$