Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Move $5$ to the left of $i$ .

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

Step 1.2

Rewrite ${(\frac{5}{3} - \frac{5 i}{3})}^{2}$ as $(\frac{5}{3} - \frac{5 i}{3}) (\frac{5}{3} - \frac{5 i}{3})$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $\frac{5}{3} \cdot \frac{5}{3}$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

Multiply $5$ by $5$ .

$\frac{25}{3 \cdot 3} + \frac{5}{3} (- \frac{5 i}{3}) - \frac{5 i}{3} \cdot \frac{5}{3} - \frac{5 i}{3} (- \frac{5 i}{3})$

Step 3.1.1.3

Multiply $3$ by $3$ .

$\frac{25}{9} + \frac{5}{3} (- \frac{5 i}{3}) - \frac{5 i}{3} \cdot \frac{5}{3} - \frac{5 i}{3} (- \frac{5 i}{3})$

Step 3.1.2

Multiply $\frac{5}{3} (- \frac{5 i}{3})$ .

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Step 3.1.2.1

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

$\frac{25}{9} - \frac{5 (5 i)}{3 \cdot 3} - \frac{5 i}{3} \cdot \frac{5}{3} - \frac{5 i}{3} (- \frac{5 i}{3})$

Step 3.1.2.2

Multiply $5$ by $5$ .

$\frac{25}{9} - \frac{25 i}{3 \cdot 3} - \frac{5 i}{3} \cdot \frac{5}{3} - \frac{5 i}{3} (- \frac{5 i}{3})$

Step 3.1.2.3

Multiply $3$ by $3$ .

$\frac{25}{9} - \frac{25 i}{9} - \frac{5 i}{3} \cdot \frac{5}{3} - \frac{5 i}{3} (- \frac{5 i}{3})$

Step 3.1.3

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

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Step 3.1.3.1

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

$\frac{25}{9} - \frac{25 i}{9} - \frac{5 (5 i)}{3 \cdot 3} - \frac{5 i}{3} (- \frac{5 i}{3})$

Step 3.1.3.2

Multiply $5$ by $5$ .

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{3 \cdot 3} - \frac{5 i}{3} (- \frac{5 i}{3})$

Step 3.1.3.3

Multiply $3$ by $3$ .

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} - \frac{5 i}{3} (- \frac{5 i}{3})$

Step 3.1.4

Multiply $- \frac{5 i}{3} (- \frac{5 i}{3})$ .

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Step 3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} + 1 \frac{5 i}{3} \frac{5 i}{3}$

Step 3.1.4.2

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

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} + \frac{5 i}{3} \cdot \frac{5 i}{3}$

Step 3.1.4.3

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

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} + \frac{5 i (5 i)}{3 \cdot 3}$

Step 3.1.4.4

Multiply $5$ by $5$ .

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} + \frac{25 i i}{3 \cdot 3}$

Step 3.1.4.5

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

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} + \frac{25 (i^{1} i)}{3 \cdot 3}$

Step 3.1.4.6

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

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} + \frac{25 (i^{1} i^{1})}{3 \cdot 3}$

Step 3.1.4.7

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

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} + \frac{25 i^{1 + 1}}{3 \cdot 3}$

Step 3.1.4.8

Add $1$ and $1$ .

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} + \frac{25 i^{2}}{3 \cdot 3}$

Step 3.1.4.9

Multiply $3$ by $3$ .

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

Step 3.1.5

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

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

Step 3.1.6

Multiply $25$ by $- 1$ .

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

Step 3.1.7

Move the negative in front of the fraction.

$\frac{25}{9} - \frac{25 i}{9} - \frac{25 i}{9} - \frac{25}{9}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{25 - 25}{9} - \frac{25 i}{9} - \frac{25 i}{9}$

Step 3.3

Subtract $25$ from $25$ .

$\frac{0}{9} - \frac{25 i}{9} - \frac{25 i}{9}$

Step 3.4

Subtract $\frac{25 i}{9}$ from $- \frac{25 i}{9}$ .

$\frac{0}{9} - 2 \frac{25 i}{9}$

Step 4

Simplify each term.

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Step 4.1

Divide $0$ by $9$ .

$0 - 2 \frac{25 i}{9}$

Step 4.2

Multiply $- 2 \frac{25 i}{9}$ .

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Step 4.2.1

Combine $- 2$ and $\frac{25 i}{9}$ .

$0 + \frac{- 2 (25 i)}{9}$

Step 4.2.2

Multiply $25$ by $- 2$ .

$0 + \frac{- 50 i}{9}$

Step 4.3

Move the negative in front of the fraction.

$0 - \frac{50 i}{9}$

Step 5

Subtract $\frac{50 i}{9}$ from $0$ .

$- \frac{50 i}{9}$