Pre-Algebra Examples

${(\frac{1}{6} + \frac{\sqrt{19}}{6} \cdot i)}^{2}$

Step 1

Combine $\frac{\sqrt{19}}{6}$ and $i$ .

${(\frac{1}{6} + \frac{\sqrt{19} i}{6})}^{2}$

Step 2

Rewrite ${(\frac{1}{6} + \frac{\sqrt{19} i}{6})}^{2}$ as $(\frac{1}{6} + \frac{\sqrt{19} i}{6}) (\frac{1}{6} + \frac{\sqrt{19} i}{6})$ .

$(\frac{1}{6} + \frac{\sqrt{19} i}{6}) (\frac{1}{6} + \frac{\sqrt{19} i}{6})$

Step 3

Expand $(\frac{1}{6} + \frac{\sqrt{19} i}{6}) (\frac{1}{6} + \frac{\sqrt{19} i}{6})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{1}{6} (\frac{1}{6} + \frac{\sqrt{19} i}{6}) + \frac{\sqrt{19} i}{6} (\frac{1}{6} + \frac{\sqrt{19} i}{6})$

Step 3.2

Apply the distributive property.

$\frac{1}{6} \cdot \frac{1}{6} + \frac{1}{6} \cdot \frac{\sqrt{19} i}{6} + \frac{\sqrt{19} i}{6} (\frac{1}{6} + \frac{\sqrt{19} i}{6})$

Step 3.3

Apply the distributive property.

$\frac{1}{6} \cdot \frac{1}{6} + \frac{1}{6} \cdot \frac{\sqrt{19} i}{6} + \frac{\sqrt{19} i}{6} \cdot \frac{1}{6} + \frac{\sqrt{19} i}{6} \cdot \frac{\sqrt{19} i}{6}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $\frac{1}{6} \cdot \frac{1}{6}$ .

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Step 4.1.1.1

Multiply $\frac{1}{6}$ by $\frac{1}{6}$ .

$\frac{1}{6 \cdot 6} + \frac{1}{6} \cdot \frac{\sqrt{19} i}{6} + \frac{\sqrt{19} i}{6} \cdot \frac{1}{6} + \frac{\sqrt{19} i}{6} \cdot \frac{\sqrt{19} i}{6}$

Step 4.1.1.2

Multiply $6$ by $6$ .

$\frac{1}{36} + \frac{1}{6} \cdot \frac{\sqrt{19} i}{6} + \frac{\sqrt{19} i}{6} \cdot \frac{1}{6} + \frac{\sqrt{19} i}{6} \cdot \frac{\sqrt{19} i}{6}$

Step 4.1.2

Multiply $\frac{1}{6} \cdot \frac{\sqrt{19} i}{6}$ .

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Step 4.1.2.1

Multiply $\frac{1}{6}$ by $\frac{\sqrt{19} i}{6}$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{6 \cdot 6} + \frac{\sqrt{19} i}{6} \cdot \frac{1}{6} + \frac{\sqrt{19} i}{6} \cdot \frac{\sqrt{19} i}{6}$

Step 4.1.2.2

Multiply $6$ by $6$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{6} \cdot \frac{1}{6} + \frac{\sqrt{19} i}{6} \cdot \frac{\sqrt{19} i}{6}$

Step 4.1.3

Multiply $\frac{\sqrt{19} i}{6} \cdot \frac{1}{6}$ .

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Step 4.1.3.1

Multiply $\frac{\sqrt{19} i}{6}$ by $\frac{1}{6}$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{6 \cdot 6} + \frac{\sqrt{19} i}{6} \cdot \frac{\sqrt{19} i}{6}$

Step 4.1.3.2

Multiply $6$ by $6$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{6} \cdot \frac{\sqrt{19} i}{6}$

Step 4.1.4

Multiply $\frac{\sqrt{19} i}{6} \cdot \frac{\sqrt{19} i}{6}$ .

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Step 4.1.4.1

Multiply $\frac{\sqrt{19} i}{6}$ by $\frac{\sqrt{19} i}{6}$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i (\sqrt{19} i)}{6 \cdot 6}$

Step 4.1.4.2

Raise $\sqrt{19}$ to the power of $1$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{1} \sqrt{19} i i}{6 \cdot 6}$

Step 4.1.4.3

Raise $\sqrt{19}$ to the power of $1$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{1} {\sqrt{19}}^{1} i i}{6 \cdot 6}$

Step 4.1.4.4

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

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{1 + 1} i i}{6 \cdot 6}$

Step 4.1.4.5

Add $1$ and $1$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{2} i i}{6 \cdot 6}$

Step 4.1.4.6

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

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{2} (i^{1} i)}{6 \cdot 6}$

Step 4.1.4.7

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

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{2} (i^{1} i^{1})}{6 \cdot 6}$

Step 4.1.4.8

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

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{2} i^{1 + 1}}{6 \cdot 6}$

Step 4.1.4.9

Add $1$ and $1$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{2} i^{2}}{6 \cdot 6}$

Step 4.1.4.10

Multiply $6$ by $6$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{\sqrt{19}}^{2} i^{2}}{36}$

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 4.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{{(19^{\frac{1}{2}})}^{2} i^{2}}{36}$

Step 4.1.5.1.2

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

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{19^{\frac{1}{2} \cdot 2} i^{2}}{36}$

Step 4.1.5.1.3

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

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{19^{\frac{2}{2}} i^{2}}{36}$

Step 4.1.5.1.4

Cancel the common factor of $2$ .

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Step 4.1.5.1.4.1

Cancel the common factor.

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{19^{\frac{2}{2}} i^{2}}{36}$

Step 4.1.5.1.4.2

Rewrite the expression.

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{19^{1} i^{2}}{36}$

Step 4.1.5.1.5

Evaluate the exponent.

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{19 i^{2}}{36}$

Step 4.1.5.2

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

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{19 \cdot - 1}{36}$

Step 4.1.6

Multiply $19$ by $- 1$ .

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} + \frac{- 19}{36}$

Step 4.1.7

Move the negative in front of the fraction.

$\frac{1}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36} - \frac{19}{36}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{1 - 19}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36}$

Step 4.3

Subtract $19$ from $1$ .

$\frac{- 18}{36} + \frac{\sqrt{19} i}{36} + \frac{\sqrt{19} i}{36}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 18}{36} + \frac{2 \sqrt{19} i}{36}$

Step 5

Simplify each term.

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Step 5.1

Cancel the common factor of $- 18$ and $36$ .

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Step 5.1.1

Factor $18$ out of $- 18$ .

$\frac{18 (- 1)}{36} + \frac{2 \sqrt{19} i}{36}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $18$ out of $36$ .

$\frac{18 \cdot - 1}{18 \cdot 2} + \frac{2 \sqrt{19} i}{36}$

Step 5.1.2.2

Cancel the common factor.

$\frac{18 \cdot - 1}{18 \cdot 2} + \frac{2 \sqrt{19} i}{36}$

Step 5.1.2.3

Rewrite the expression.

$\frac{- 1}{2} + \frac{2 \sqrt{19} i}{36}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{1}{2} + \frac{2 \sqrt{19} i}{36}$

Step 5.3

Cancel the common factor of $2$ and $36$ .

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Step 5.3.1

Factor $2$ out of $2 \sqrt{19} i$ .

$- \frac{1}{2} + \frac{2 (\sqrt{19} i)}{36}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $2$ out of $36$ .

$- \frac{1}{2} + \frac{2 (\sqrt{19} i)}{2 \cdot 18}$

Step 5.3.2.2

Cancel the common factor.

$- \frac{1}{2} + \frac{2 (\sqrt{19} i)}{2 \cdot 18}$

Step 5.3.2.3

Rewrite the expression.

$- \frac{1}{2} + \frac{\sqrt{19} i}{18}$