Precalculus Examples

${(\frac{1}{3} + \frac{\sqrt{7}}{6} \cdot i)}^{2}$

Step 1

Simplify terms.

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Step 1.1

Combine $\frac{\sqrt{7}}{6}$ and $i$ .

${(\frac{1}{3} + \frac{\sqrt{7} i}{6})}^{2}$

Step 1.2

Rewrite ${(\frac{1}{3} + \frac{\sqrt{7} i}{6})}^{2}$ as $(\frac{1}{3} + \frac{\sqrt{7} i}{6}) (\frac{1}{3} + \frac{\sqrt{7} i}{6})$ .

$(\frac{1}{3} + \frac{\sqrt{7} i}{6}) (\frac{1}{3} + \frac{\sqrt{7} i}{6})$

Step 2

Expand $(\frac{1}{3} + \frac{\sqrt{7} i}{6}) (\frac{1}{3} + \frac{\sqrt{7} i}{6})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{1}{3} (\frac{1}{3} + \frac{\sqrt{7} i}{6}) + \frac{\sqrt{7} i}{6} (\frac{1}{3} + \frac{\sqrt{7} i}{6})$

Step 2.2

Apply the distributive property.

$\frac{1}{3} \cdot \frac{1}{3} + \frac{1}{3} \cdot \frac{\sqrt{7} i}{6} + \frac{\sqrt{7} i}{6} (\frac{1}{3} + \frac{\sqrt{7} i}{6})$

Step 2.3

Apply the distributive property.

$\frac{1}{3} \cdot \frac{1}{3} + \frac{1}{3} \cdot \frac{\sqrt{7} i}{6} + \frac{\sqrt{7} i}{6} \cdot \frac{1}{3} + \frac{\sqrt{7} i}{6} \cdot \frac{\sqrt{7} i}{6}$

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 $\frac{1}{3} \cdot \frac{1}{3}$ .

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Step 3.1.1.1

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

$\frac{1}{3 \cdot 3} + \frac{1}{3} \cdot \frac{\sqrt{7} i}{6} + \frac{\sqrt{7} i}{6} \cdot \frac{1}{3} + \frac{\sqrt{7} i}{6} \cdot \frac{\sqrt{7} i}{6}$

Step 3.1.1.2

Multiply $3$ by $3$ .

$\frac{1}{9} + \frac{1}{3} \cdot \frac{\sqrt{7} i}{6} + \frac{\sqrt{7} i}{6} \cdot \frac{1}{3} + \frac{\sqrt{7} i}{6} \cdot \frac{\sqrt{7} i}{6}$

Step 3.1.2

Multiply $\frac{1}{3} \cdot \frac{\sqrt{7} i}{6}$ .

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Step 3.1.2.1

Multiply $\frac{1}{3}$ by $\frac{\sqrt{7} i}{6}$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{3 \cdot 6} + \frac{\sqrt{7} i}{6} \cdot \frac{1}{3} + \frac{\sqrt{7} i}{6} \cdot \frac{\sqrt{7} i}{6}$

Step 3.1.2.2

Multiply $3$ by $6$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{6} \cdot \frac{1}{3} + \frac{\sqrt{7} i}{6} \cdot \frac{\sqrt{7} i}{6}$

Step 3.1.3

Multiply $\frac{\sqrt{7} i}{6} \cdot \frac{1}{3}$ .

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Step 3.1.3.1

Multiply $\frac{\sqrt{7} i}{6}$ by $\frac{1}{3}$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{6 \cdot 3} + \frac{\sqrt{7} i}{6} \cdot \frac{\sqrt{7} i}{6}$

Step 3.1.3.2

Multiply $6$ by $3$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{6} \cdot \frac{\sqrt{7} i}{6}$

Step 3.1.4

Multiply $\frac{\sqrt{7} i}{6} \cdot \frac{\sqrt{7} i}{6}$ .

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Step 3.1.4.1

Multiply $\frac{\sqrt{7} i}{6}$ by $\frac{\sqrt{7} i}{6}$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i (\sqrt{7} i)}{6 \cdot 6}$

Step 3.1.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{1} \sqrt{7} i i}{6 \cdot 6}$

Step 3.1.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{1} {\sqrt{7}}^{1} i i}{6 \cdot 6}$

Step 3.1.4.4

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

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{1 + 1} i i}{6 \cdot 6}$

Step 3.1.4.5

Add $1$ and $1$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{2} i i}{6 \cdot 6}$

Step 3.1.4.6

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

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{2} (i^{1} i)}{6 \cdot 6}$

Step 3.1.4.7

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

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{2} (i^{1} i^{1})}{6 \cdot 6}$

Step 3.1.4.8

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

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{2} i^{1 + 1}}{6 \cdot 6}$

Step 3.1.4.9

Add $1$ and $1$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{2} i^{2}}{6 \cdot 6}$

Step 3.1.4.10

Multiply $6$ by $6$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{\sqrt{7}}^{2} i^{2}}{36}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{{(7^{\frac{1}{2}})}^{2} i^{2}}{36}$

Step 3.1.5.1.2

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

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{7^{\frac{1}{2} \cdot 2} i^{2}}{36}$

Step 3.1.5.1.3

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

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{7^{\frac{2}{2}} i^{2}}{36}$

Step 3.1.5.1.4

Cancel the common factor of $2$ .

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Step 3.1.5.1.4.1

Cancel the common factor.

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{7^{\frac{2}{2}} i^{2}}{36}$

Step 3.1.5.1.4.2

Rewrite the expression.

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{7^{1} i^{2}}{36}$

Step 3.1.5.1.5

Evaluate the exponent.

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{7 i^{2}}{36}$

Step 3.1.5.2

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

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{7 \cdot - 1}{36}$

Step 3.1.6

Multiply $7$ by $- 1$ .

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} + \frac{- 7}{36}$

Step 3.1.7

Move the negative in front of the fraction.

$\frac{1}{9} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18} - \frac{7}{36}$

Step 3.2

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{9} \cdot \frac{4}{4} - \frac{7}{36} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18}$

Step 3.3

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

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Step 3.3.1

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$\frac{4}{9 \cdot 4} - \frac{7}{36} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18}$

Step 3.3.2

Multiply $9$ by $4$ .

$\frac{4}{36} - \frac{7}{36} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{4 - 7}{36} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18}$

Step 3.5

Subtract $7$ from $4$ .

$\frac{- 3}{36} + \frac{\sqrt{7} i}{18} + \frac{\sqrt{7} i}{18}$

Step 3.6

Combine the numerators over the common denominator.

$\frac{- 3}{36} + \frac{2 \sqrt{7} i}{18}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $- 3$ and $36$ .

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Step 4.1.1

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{36} + \frac{2 \sqrt{7} i}{18}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $3$ out of $36$ .

$\frac{3 \cdot - 1}{3 \cdot 12} + \frac{2 \sqrt{7} i}{18}$

Step 4.1.2.2

Cancel the common factor.

$\frac{3 \cdot - 1}{3 \cdot 12} + \frac{2 \sqrt{7} i}{18}$

Step 4.1.2.3

Rewrite the expression.

$\frac{- 1}{12} + \frac{2 \sqrt{7} i}{18}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{1}{12} + \frac{2 \sqrt{7} i}{18}$

Step 4.3

Cancel the common factor of $2$ and $18$ .

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Step 4.3.1

Factor $2$ out of $2 \sqrt{7} i$ .

$- \frac{1}{12} + \frac{2 (\sqrt{7} i)}{18}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $18$ .

$- \frac{1}{12} + \frac{2 (\sqrt{7} i)}{2 \cdot 9}$

Step 4.3.2.2

Cancel the common factor.

$- \frac{1}{12} + \frac{2 (\sqrt{7} i)}{2 \cdot 9}$

Step 4.3.2.3

Rewrite the expression.

$- \frac{1}{12} + \frac{\sqrt{7} i}{9}$