Precalculus Examples

$(\frac{1}{3} - \frac{1}{5} \cdot i) (\frac{1}{3} + \frac{1}{5} \cdot i)$

Step 1

Simplify terms.

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Step 1.1

Combine $i$ and $\frac{1}{5}$ .

$(\frac{1}{3} - \frac{i}{5}) (\frac{1}{3} + \frac{1}{5} \cdot i)$

Step 1.2

Combine $\frac{1}{5}$ and $i$ .

$(\frac{1}{3} - \frac{i}{5}) (\frac{1}{3} + \frac{i}{5})$

Step 2

Expand $(\frac{1}{3} - \frac{i}{5}) (\frac{1}{3} + \frac{i}{5})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{1}{3} (\frac{1}{3} + \frac{i}{5}) - \frac{i}{5} (\frac{1}{3} + \frac{i}{5})$

Step 2.2

Apply the distributive property.

$\frac{1}{3} \cdot \frac{1}{3} + \frac{1}{3} \cdot \frac{i}{5} - \frac{i}{5} (\frac{1}{3} + \frac{i}{5})$

Step 2.3

Apply the distributive property.

$\frac{1}{3} \cdot \frac{1}{3} + \frac{1}{3} \cdot \frac{i}{5} - \frac{i}{5} \cdot \frac{1}{3} - \frac{i}{5} \cdot \frac{i}{5}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\frac{1}{3} \cdot \frac{1}{3}$ .

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Step 3.1.1.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$\frac{1}{3 \cdot 3} + \frac{1}{3} \cdot \frac{i}{5} - \frac{i}{5} \cdot \frac{1}{3} - \frac{i}{5} \cdot \frac{i}{5}$

Step 3.1.1.2

Multiply $3$ by $3$ .

$\frac{1}{9} + \frac{1}{3} \cdot \frac{i}{5} - \frac{i}{5} \cdot \frac{1}{3} - \frac{i}{5} \cdot \frac{i}{5}$

Step 3.1.2

Multiply $\frac{1}{3} \cdot \frac{i}{5}$ .

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Step 3.1.2.1

Multiply $\frac{1}{3}$ by $\frac{i}{5}$ .

$\frac{1}{9} + \frac{i}{3 \cdot 5} - \frac{i}{5} \cdot \frac{1}{3} - \frac{i}{5} \cdot \frac{i}{5}$

Step 3.1.2.2

Multiply $3$ by $5$ .

$\frac{1}{9} + \frac{i}{15} - \frac{i}{5} \cdot \frac{1}{3} - \frac{i}{5} \cdot \frac{i}{5}$

Step 3.1.3

Multiply $- \frac{i}{5} \cdot \frac{1}{3}$ .

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Step 3.1.3.1

Multiply $\frac{1}{3}$ by $\frac{i}{5}$ .

$\frac{1}{9} + \frac{i}{15} - \frac{i}{3 \cdot 5} - \frac{i}{5} \cdot \frac{i}{5}$

Step 3.1.3.2

Multiply $3$ by $5$ .

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - \frac{i}{5} \cdot \frac{i}{5}$

Step 3.1.4

Multiply $- \frac{i}{5} \cdot \frac{i}{5}$ .

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Step 3.1.4.1

Multiply $\frac{i}{5}$ by $\frac{i}{5}$ .

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - \frac{i i}{5 \cdot 5}$

Step 3.1.4.2

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

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - \frac{i^{1} i}{5 \cdot 5}$

Step 3.1.4.3

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

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - \frac{i^{1} i^{1}}{5 \cdot 5}$

Step 3.1.4.4

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

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - \frac{i^{1 + 1}}{5 \cdot 5}$

Step 3.1.4.5

Add $1$ and $1$ .

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - \frac{i^{2}}{5 \cdot 5}$

Step 3.1.4.6

Multiply $5$ by $5$ .

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - \frac{i^{2}}{25}$

Step 3.1.5

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

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - \frac{- 1}{25}$

Step 3.1.6

Move the negative in front of the fraction.

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} - - \frac{1}{25}$

Step 3.1.7

Multiply $- - \frac{1}{25}$ .

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Step 3.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} + 1 (\frac{1}{25})$

Step 3.1.7.2

Multiply $\frac{1}{25}$ by $1$ .

$\frac{1}{9} + \frac{i}{15} - \frac{i}{15} + \frac{1}{25}$

Step 3.2

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{1}{9} \cdot \frac{25}{25} + \frac{1}{25} + \frac{i}{15} - \frac{i}{15}$

Step 3.3

To write $\frac{1}{25}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{1}{9} \cdot \frac{25}{25} + \frac{1}{25} \cdot \frac{9}{9} + \frac{i}{15} - \frac{i}{15}$

Step 3.4

Write each expression with a common denominator of $225$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{1}{9}$ by $\frac{25}{25}$ .

$\frac{25}{9 \cdot 25} + \frac{1}{25} \cdot \frac{9}{9} + \frac{i}{15} - \frac{i}{15}$

Step 3.4.2

Multiply $9$ by $25$ .

$\frac{25}{225} + \frac{1}{25} \cdot \frac{9}{9} + \frac{i}{15} - \frac{i}{15}$

Step 3.4.3

Multiply $\frac{1}{25}$ by $\frac{9}{9}$ .

$\frac{25}{225} + \frac{9}{25 \cdot 9} + \frac{i}{15} - \frac{i}{15}$

Step 3.4.4

Multiply $25$ by $9$ .

$\frac{25}{225} + \frac{9}{225} + \frac{i}{15} - \frac{i}{15}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{25 + 9}{225} + \frac{i}{15} - \frac{i}{15}$

Step 3.6

Add $25$ and $9$ .

$\frac{34}{225} + \frac{i}{15} - \frac{i}{15}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{34}{225} + \frac{0}{15}$

Step 3.8

To write $\frac{0}{15}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{34}{225} + \frac{0}{15} \cdot \frac{15}{15}$

Step 3.9

Write each expression with a common denominator of $225$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.9.1

Multiply $\frac{0}{15}$ by $\frac{15}{15}$ .

$\frac{34}{225} + \frac{0 \cdot 15}{15 \cdot 15}$

Step 3.9.2

Multiply $15$ by $15$ .

$\frac{34}{225} + \frac{0 \cdot 15}{225}$

Step 3.10

Combine the numerators over the common denominator.

$\frac{34 + 0 \cdot 15}{225}$

Step 3.11

Simplify the numerator.

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Step 3.11.1

Multiply $0$ by $15$ .

$\frac{34 + 0}{225}$

Step 3.11.2

Add $34$ and $0$ .

$\frac{34}{225}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{34}{225}$

Decimal Form:

$0.15\overline{1}$