Algebra Examples

$(\frac{2}{3} + \frac{1}{2} i) (12 - \frac{1}{3} i)$

Step 1

Simplify terms.

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Step 1.1

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

$(\frac{2}{3} + \frac{i}{2}) (12 - \frac{1}{3} \cdot i)$

Step 1.2

Combine $i$ and $\frac{1}{3}$ .

$(\frac{2}{3} + \frac{i}{2}) (12 - \frac{i}{3})$

Step 2

Expand $(\frac{2}{3} + \frac{i}{2}) (12 - \frac{i}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{2}{3} (12 - \frac{i}{3}) + \frac{i}{2} (12 - \frac{i}{3})$

Step 2.2

Apply the distributive property.

$\frac{2}{3} \cdot 12 + \frac{2}{3} (- \frac{i}{3}) + \frac{i}{2} (12 - \frac{i}{3})$

Step 2.3

Apply the distributive property.

$\frac{2}{3} \cdot 12 + \frac{2}{3} (- \frac{i}{3}) + \frac{i}{2} \cdot 12 + \frac{i}{2} (- \frac{i}{3})$

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

Cancel the common factor of $3$ .

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Step 3.1.1.1

Factor $3$ out of $12$ .

$\frac{2}{3} \cdot (3 (4)) + \frac{2}{3} (- \frac{i}{3}) + \frac{i}{2} \cdot 12 + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.1.2

Cancel the common factor.

$\frac{2}{3} \cdot (3 \cdot 4) + \frac{2}{3} (- \frac{i}{3}) + \frac{i}{2} \cdot 12 + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.1.3

Rewrite the expression.

$2 \cdot 4 + \frac{2}{3} (- \frac{i}{3}) + \frac{i}{2} \cdot 12 + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.2

Multiply $2$ by $4$ .

$8 + \frac{2}{3} (- \frac{i}{3}) + \frac{i}{2} \cdot 12 + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.3

Multiply $\frac{2}{3} (- \frac{i}{3})$ .

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Step 3.1.3.1

Multiply $\frac{2}{3}$ by $\frac{i}{3}$ .

$8 - \frac{2 i}{3 \cdot 3} + \frac{i}{2} \cdot 12 + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.3.2

Multiply $3$ by $3$ .

$8 - \frac{2 i}{9} + \frac{i}{2} \cdot 12 + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Factor $2$ out of $12$ .

$8 - \frac{2 i}{9} + \frac{i}{2} \cdot (2 (6)) + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.4.2

Cancel the common factor.

$8 - \frac{2 i}{9} + \frac{i}{2} \cdot (2 \cdot 6) + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.4.3

Rewrite the expression.

$8 - \frac{2 i}{9} + i \cdot 6 + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.5

Move $6$ to the left of $i$ .

$8 - \frac{2 i}{9} + 6 \cdot i + \frac{i}{2} (- \frac{i}{3})$

Step 3.1.6

Multiply $\frac{i}{2} (- \frac{i}{3})$ .

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Step 3.1.6.1

Multiply $\frac{i}{2}$ by $\frac{i}{3}$ .

$8 - \frac{2 i}{9} + 6 i - \frac{i i}{2 \cdot 3}$

Step 3.1.6.2

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

$8 - \frac{2 i}{9} + 6 i - \frac{i^{1} i}{2 \cdot 3}$

Step 3.1.6.3

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

$8 - \frac{2 i}{9} + 6 i - \frac{i^{1} i^{1}}{2 \cdot 3}$

Step 3.1.6.4

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

$8 - \frac{2 i}{9} + 6 i - \frac{i^{1 + 1}}{2 \cdot 3}$

Step 3.1.6.5

Add $1$ and $1$ .

$8 - \frac{2 i}{9} + 6 i - \frac{i^{2}}{2 \cdot 3}$

Step 3.1.6.6

Multiply $2$ by $3$ .

$8 - \frac{2 i}{9} + 6 i - \frac{i^{2}}{6}$

Step 3.1.7

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

$8 - \frac{2 i}{9} + 6 i - \frac{- 1}{6}$

Step 3.1.8

Move the negative in front of the fraction.

$8 - \frac{2 i}{9} + 6 i - - \frac{1}{6}$

Step 3.1.9

Multiply $- - \frac{1}{6}$ .

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Step 3.1.9.1

Multiply $- 1$ by $- 1$ .

$8 - \frac{2 i}{9} + 6 i + 1 (\frac{1}{6})$

Step 3.1.9.2

Multiply $\frac{1}{6}$ by $1$ .

$8 - \frac{2 i}{9} + 6 i + \frac{1}{6}$

Step 3.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$8 \cdot \frac{6}{6} + \frac{1}{6} - \frac{2 i}{9} + 6 i$

Step 3.3

Combine $8$ and $\frac{6}{6}$ .

$\frac{8 \cdot 6}{6} + \frac{1}{6} - \frac{2 i}{9} + 6 i$

Step 3.4

Combine the numerators over the common denominator.

$\frac{8 \cdot 6 + 1}{6} - \frac{2 i}{9} + 6 i$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $8$ by $6$ .

$\frac{48 + 1}{6} - \frac{2 i}{9} + 6 i$

Step 3.5.2

Add $48$ and $1$ .

$\frac{49}{6} - \frac{2 i}{9} + 6 i$

Step 3.6

To write $6 i$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{49}{6} - \frac{2 i}{9} + 6 i \cdot \frac{9}{9}$

Step 3.7

Combine $6 i$ and $\frac{9}{9}$ .

$\frac{49}{6} - \frac{2 i}{9} + \frac{6 i \cdot 9}{9}$

Step 3.8

Combine the numerators over the common denominator.

$\frac{49}{6} + \frac{52 i}{9}$