Algebra Examples

${(\frac{2}{3} + \frac{2}{3} i)}^{2}$

Step 1

Simplify terms.

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Step 1.1

Combine $\frac{2}{3}$ and $i$ .

${(\frac{2}{3} + \frac{2 i}{3})}^{2}$

Step 1.2

Rewrite ${(\frac{2}{3} + \frac{2 i}{3})}^{2}$ as $(\frac{2}{3} + \frac{2 i}{3}) (\frac{2}{3} + \frac{2 i}{3})$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$\frac{2}{3} \cdot \frac{2}{3} + \frac{2}{3} \cdot \frac{2 i}{3} + \frac{2 i}{3} \cdot \frac{2}{3} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

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

Multiply $\frac{2}{3} \cdot \frac{2}{3}$ .

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Step 3.1.1.1

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

$\frac{2 \cdot 2}{3 \cdot 3} + \frac{2}{3} \cdot \frac{2 i}{3} + \frac{2 i}{3} \cdot \frac{2}{3} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

Step 3.1.1.2

Multiply $2$ by $2$ .

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

Step 3.1.1.3

Multiply $3$ by $3$ .

$\frac{4}{9} + \frac{2}{3} \cdot \frac{2 i}{3} + \frac{2 i}{3} \cdot \frac{2}{3} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

Step 3.1.2

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

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Step 3.1.2.1

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

$\frac{4}{9} + \frac{2 (2 i)}{3 \cdot 3} + \frac{2 i}{3} \cdot \frac{2}{3} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

Step 3.1.2.2

Multiply $2$ by $2$ .

$\frac{4}{9} + \frac{4 i}{3 \cdot 3} + \frac{2 i}{3} \cdot \frac{2}{3} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

Step 3.1.2.3

Multiply $3$ by $3$ .

$\frac{4}{9} + \frac{4 i}{9} + \frac{2 i}{3} \cdot \frac{2}{3} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

Step 3.1.3

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

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Step 3.1.3.1

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

$\frac{4}{9} + \frac{4 i}{9} + \frac{2 i \cdot 2}{3 \cdot 3} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

Step 3.1.3.2

Multiply $2$ by $2$ .

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{3 \cdot 3} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

Step 3.1.3.3

Multiply $3$ by $3$ .

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{9} + \frac{2 i}{3} \cdot \frac{2 i}{3}$

Step 3.1.4

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

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Step 3.1.4.1

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

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{9} + \frac{2 i (2 i)}{3 \cdot 3}$

Step 3.1.4.2

Multiply $2$ by $2$ .

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{9} + \frac{4 i i}{3 \cdot 3}$

Step 3.1.4.3

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

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{9} + \frac{4 (i^{1} i)}{3 \cdot 3}$

Step 3.1.4.4

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

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{9} + \frac{4 (i^{1} i^{1})}{3 \cdot 3}$

Step 3.1.4.5

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

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{9} + \frac{4 i^{1 + 1}}{3 \cdot 3}$

Step 3.1.4.6

Add $1$ and $1$ .

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{9} + \frac{4 i^{2}}{3 \cdot 3}$

Step 3.1.4.7

Multiply $3$ by $3$ .

$\frac{4}{9} + \frac{4 i}{9} + \frac{4 i}{9} + \frac{4 i^{2}}{9}$

Step 3.1.5

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

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

Step 3.1.6

Multiply $4$ by $- 1$ .

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

Step 3.1.7

Move the negative in front of the fraction.

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Subtract $4$ from $4$ .

$\frac{0}{9} + \frac{4 i}{9} + \frac{4 i}{9}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{0}{9} + \frac{8 i}{9}$

Step 4

Divide $0$ by $9$ .

$0 + \frac{8 i}{9}$

Step 5

Add $0$ and $\frac{8 i}{9}$ .

$\frac{8 i}{9}$