Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

Multiply $3$ by $3$ .

$\frac{9}{5 \cdot 5} + \frac{3}{5} \cdot \frac{3 i}{2} + \frac{3 i}{2} \cdot \frac{3}{5} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.1.3

Multiply $5$ by $5$ .

$\frac{9}{25} + \frac{3}{5} \cdot \frac{3 i}{2} + \frac{3 i}{2} \cdot \frac{3}{5} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.2

Multiply $\frac{3}{5} \cdot \frac{3 i}{2}$ .

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Step 3.1.2.1

Multiply $\frac{3}{5}$ by $\frac{3 i}{2}$ .

$\frac{9}{25} + \frac{3 (3 i)}{5 \cdot 2} + \frac{3 i}{2} \cdot \frac{3}{5} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.2.2

Multiply $3$ by $3$ .

$\frac{9}{25} + \frac{9 i}{5 \cdot 2} + \frac{3 i}{2} \cdot \frac{3}{5} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.2.3

Multiply $5$ by $2$ .

$\frac{9}{25} + \frac{9 i}{10} + \frac{3 i}{2} \cdot \frac{3}{5} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.3

Multiply $\frac{3 i}{2} \cdot \frac{3}{5}$ .

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Step 3.1.3.1

Multiply $\frac{3 i}{2}$ by $\frac{3}{5}$ .

$\frac{9}{25} + \frac{9 i}{10} + \frac{3 i \cdot 3}{2 \cdot 5} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.3.2

Multiply $3$ by $3$ .

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{2 \cdot 5} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.3.3

Multiply $2$ by $5$ .

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{3 i}{2} \cdot \frac{3 i}{2}$

Step 3.1.4

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

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Step 3.1.4.1

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

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{3 i (3 i)}{2 \cdot 2}$

Step 3.1.4.2

Multiply $3$ by $3$ .

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{9 i i}{2 \cdot 2}$

Step 3.1.4.3

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

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{9 (i^{1} i)}{2 \cdot 2}$

Step 3.1.4.4

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

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{9 (i^{1} i^{1})}{2 \cdot 2}$

Step 3.1.4.5

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

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{9 i^{1 + 1}}{2 \cdot 2}$

Step 3.1.4.6

Add $1$ and $1$ .

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{9 i^{2}}{2 \cdot 2}$

Step 3.1.4.7

Multiply $2$ by $2$ .

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{9 i^{2}}{4}$

Step 3.1.5

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

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{9 \cdot - 1}{4}$

Step 3.1.6

Multiply $9$ by $- 1$ .

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} + \frac{- 9}{4}$

Step 3.1.7

Move the negative in front of the fraction.

$\frac{9}{25} + \frac{9 i}{10} + \frac{9 i}{10} - \frac{9}{4}$

Step 3.2

To write $\frac{9}{25}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{9}{25} \cdot \frac{4}{4} - \frac{9}{4} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.3

To write $- \frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{9}{25} \cdot \frac{4}{4} - \frac{9}{4} \cdot \frac{25}{25} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.4

Write each expression with a common denominator of $100$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{9}{25}$ by $\frac{4}{4}$ .

$\frac{9 \cdot 4}{25 \cdot 4} - \frac{9}{4} \cdot \frac{25}{25} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.4.2

Multiply $25$ by $4$ .

$\frac{9 \cdot 4}{100} - \frac{9}{4} \cdot \frac{25}{25} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.4.3

Multiply $\frac{9}{4}$ by $\frac{25}{25}$ .

$\frac{9 \cdot 4}{100} - \frac{9 \cdot 25}{4 \cdot 25} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.4.4

Multiply $4$ by $25$ .

$\frac{9 \cdot 4}{100} - \frac{9 \cdot 25}{100} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{9 \cdot 4 - 9 \cdot 25}{100} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $9$ by $4$ .

$\frac{36 - 9 \cdot 25}{100} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.6.2

Multiply $- 9$ by $25$ .

$\frac{36 - 225}{100} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.6.3

Subtract $225$ from $36$ .

$\frac{- 189}{100} + \frac{9 i}{10} + \frac{9 i}{10}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{- 189}{100} + \frac{18 i}{10}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{189}{100} + \frac{18 i}{10}$

Step 4.2

Cancel the common factor of $18$ and $10$ .

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Step 4.2.1

Factor $2$ out of $18 i$ .

$- \frac{189}{100} + \frac{2 (9 i)}{10}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $10$ .

$- \frac{189}{100} + \frac{2 (9 i)}{2 (5)}$

Step 4.2.2.2

Cancel the common factor.

$- \frac{189}{100} + \frac{2 (9 i)}{2 \cdot 5}$

Step 4.2.2.3

Rewrite the expression.

$- \frac{189}{100} + \frac{9 i}{5}$