Algebra Examples

$(\frac{6}{5} + \frac{3}{5} \cdot i) (\frac{2}{3} - \frac{1}{3} \cdot i)$

Step 1

Simplify terms.

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Step 1.1

Combine $\frac{3}{5}$ and $i$ .

$(\frac{6}{5} + \frac{3 i}{5}) (\frac{2}{3} - \frac{1}{3} \cdot i)$

Step 1.2

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

$(\frac{6}{5} + \frac{3 i}{5}) (\frac{2}{3} - \frac{i}{3})$

Step 2

Expand $(\frac{6}{5} + \frac{3 i}{5}) (\frac{2}{3} - \frac{i}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{6}{5} (\frac{2}{3} - \frac{i}{3}) + \frac{3 i}{5} (\frac{2}{3} - \frac{i}{3})$

Step 2.2

Apply the distributive property.

$\frac{6}{5} \cdot \frac{2}{3} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} (\frac{2}{3} - \frac{i}{3})$

Step 2.3

Apply the distributive property.

$\frac{6}{5} \cdot \frac{2}{3} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

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 $6$ .

$\frac{3 (2)}{5} \cdot \frac{2}{3} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.1.2

Cancel the common factor.

$\frac{3 \cdot 2}{5} \cdot \frac{2}{3} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.1.3

Rewrite the expression.

$\frac{2}{5} \cdot 2 + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.2

Combine $\frac{2}{5}$ and $2$ .

$\frac{2 \cdot 2}{5} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.3

Multiply $2$ by $2$ .

$\frac{4}{5} + \frac{6}{5} (- \frac{i}{3}) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.4

Cancel the common factor of $3$ .

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Step 3.1.4.1

Move the leading negative in $- \frac{i}{3}$ into the numerator.

$\frac{4}{5} + \frac{6}{5} \cdot \frac{- i}{3} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.4.2

Factor $3$ out of $6$ .

$\frac{4}{5} + \frac{3 (2)}{5} \cdot \frac{- i}{3} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.4.3

Cancel the common factor.

$\frac{4}{5} + \frac{3 \cdot 2}{5} \cdot \frac{- i}{3} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.4.4

Rewrite the expression.

$\frac{4}{5} + \frac{2}{5} (- i) + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.5

Combine $\frac{2}{5}$ and $i$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.6

Cancel the common factor of $3$ .

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Step 3.1.6.1

Factor $3$ out of $3 i$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{3 (i)}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.6.2

Cancel the common factor.

$\frac{4}{5} - \frac{2 i}{5} + \frac{3 i}{5} \cdot \frac{2}{3} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.6.3

Rewrite the expression.

$\frac{4}{5} - \frac{2 i}{5} + \frac{i}{5} \cdot 2 + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.7

Combine $\frac{i}{5}$ and $2$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{i \cdot 2}{5} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.8

Move $2$ to the left of $i$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{3 i}{5} (- \frac{i}{3})$

Step 3.1.9

Cancel the common factor of $3$ .

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Step 3.1.9.1

Move the leading negative in $- \frac{i}{3}$ into the numerator.

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{3 i}{5} \cdot \frac{- i}{3}$

Step 3.1.9.2

Factor $3$ out of $3 i$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{3 (i)}{5} \cdot \frac{- i}{3}$

Step 3.1.9.3

Cancel the common factor.

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{3 i}{5} \cdot \frac{- i}{3}$

Step 3.1.9.4

Rewrite the expression.

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{i}{5} (- i)$

Step 3.1.10

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i i}{5}$

Step 3.1.11

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i^{1} i}{5}$

Step 3.1.12

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i^{1} i^{1}}{5}$

Step 3.1.13

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i^{1 + 1}}{5}$

Step 3.1.14

Add $1$ and $1$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{i^{2}}{5}$

Step 3.1.15

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - \frac{- 1}{5}$

Step 3.1.16

Move the negative in front of the fraction.

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} - - \frac{1}{5}$

Step 3.1.17

Multiply $- - \frac{1}{5}$ .

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Step 3.1.17.1

Multiply $- 1$ by $- 1$ .

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + 1 (\frac{1}{5})$

Step 3.1.17.2

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

$\frac{4}{5} - \frac{2 i}{5} + \frac{2 i}{5} + \frac{1}{5}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{4 + 1}{5} - \frac{2 i}{5} + \frac{2 i}{5}$

Step 3.3

Add $4$ and $1$ .

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

Step 3.4

Add $- \frac{2 i}{5}$ and $\frac{2 i}{5}$ .

$\frac{5}{5} + 0$

Step 3.5

Add $\frac{5}{5}$ and $0$ .

$\frac{5}{5}$

Step 4

Divide $5$ by $5$ .

$1$