Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

Multiply $2$ by $2$ .

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

Step 3.1.1.3

Multiply $3$ by $3$ .

$\frac{4}{9} + \frac{2}{3} \cdot \frac{\sqrt{5} i}{2} + \frac{\sqrt{5} i}{2} \cdot \frac{2}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Cancel the common factor.

$\frac{4}{9} + \frac{2}{3} \cdot \frac{\sqrt{5} i}{2} + \frac{\sqrt{5} i}{2} \cdot \frac{2}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.2.2

Rewrite the expression.

$\frac{4}{9} + \frac{1}{3} (\sqrt{5} i) + \frac{\sqrt{5} i}{2} \cdot \frac{2}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.3

Combine $\sqrt{5}$ and $\frac{1}{3}$ .

$\frac{4}{9} + \frac{\sqrt{5}}{3} i + \frac{\sqrt{5} i}{2} \cdot \frac{2}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.4

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{2}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.5

Cancel the common factor of $2$ .

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Step 3.1.5.1

Cancel the common factor.

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{2}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.5.2

Rewrite the expression.

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \sqrt{5} i \frac{1}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.6

Combine $\frac{1}{3}$ and $\sqrt{5}$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5}}{3} i + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.7

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{2} \cdot \frac{\sqrt{5} i}{2}$

Step 3.1.8

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

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Step 3.1.8.1

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i (\sqrt{5} i)}{2 \cdot 2}$

Step 3.1.8.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{1} \sqrt{5} i i}{2 \cdot 2}$

Step 3.1.8.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{1} {\sqrt{5}}^{1} i i}{2 \cdot 2}$

Step 3.1.8.4

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{1 + 1} i i}{2 \cdot 2}$

Step 3.1.8.5

Add $1$ and $1$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{2} i i}{2 \cdot 2}$

Step 3.1.8.6

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{2} (i^{1} i)}{2 \cdot 2}$

Step 3.1.8.7

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{2} (i^{1} i^{1})}{2 \cdot 2}$

Step 3.1.8.8

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{2} i^{1 + 1}}{2 \cdot 2}$

Step 3.1.8.9

Add $1$ and $1$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{2} i^{2}}{2 \cdot 2}$

Step 3.1.8.10

Multiply $2$ by $2$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{\sqrt{5}}^{2} i^{2}}{4}$

Step 3.1.9

Simplify the numerator.

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Step 3.1.9.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.1.9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{{(5^{\frac{1}{2}})}^{2} i^{2}}{4}$

Step 3.1.9.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{5^{\frac{1}{2} \cdot 2} i^{2}}{4}$

Step 3.1.9.1.3

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{5^{\frac{2}{2}} i^{2}}{4}$

Step 3.1.9.1.4

Cancel the common factor of $2$ .

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Step 3.1.9.1.4.1

Cancel the common factor.

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{5^{\frac{2}{2}} i^{2}}{4}$

Step 3.1.9.1.4.2

Rewrite the expression.

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{5^{1} i^{2}}{4}$

Step 3.1.9.1.5

Evaluate the exponent.

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{5 i^{2}}{4}$

Step 3.1.9.2

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

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{5 \cdot - 1}{4}$

Step 3.1.10

Multiply $5$ by $- 1$ .

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} + \frac{- 5}{4}$

Step 3.1.11

Move the negative in front of the fraction.

$\frac{4}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3} - \frac{5}{4}$

Step 3.2

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

$\frac{4}{9} \cdot \frac{4}{4} - \frac{5}{4} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.3

To write $- \frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{4}{9} \cdot \frac{4}{4} - \frac{5}{4} \cdot \frac{9}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.4

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

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

$\frac{4 \cdot 4}{9 \cdot 4} - \frac{5}{4} \cdot \frac{9}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.4.2

Multiply $9$ by $4$ .

$\frac{4 \cdot 4}{36} - \frac{5}{4} \cdot \frac{9}{9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.4.3

Multiply $\frac{5}{4}$ by $\frac{9}{9}$ .

$\frac{4 \cdot 4}{36} - \frac{5 \cdot 9}{4 \cdot 9} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.4.4

Multiply $4$ by $9$ .

$\frac{4 \cdot 4}{36} - \frac{5 \cdot 9}{36} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{4 \cdot 4 - 5 \cdot 9}{36} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $4$ by $4$ .

$\frac{16 - 5 \cdot 9}{36} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.6.2

Multiply $- 5$ by $9$ .

$\frac{16 - 45}{36} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.6.3

Subtract $45$ from $16$ .

$\frac{- 29}{36} + \frac{\sqrt{5} i}{3} + \frac{\sqrt{5} i}{3}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{- 29}{36} + \frac{2 \sqrt{5} i}{3}$

Step 4

Move the negative in front of the fraction.

$- \frac{29}{36} + \frac{2 \sqrt{5} i}{3}$