Algebra Examples

${(\frac{1}{8} + \frac{\sqrt{17}}{8} i)}^{2}$

Step 1

Combine fractions.

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Step 1.1

Combine $\frac{\sqrt{17}}{8}$ and $i$ .

${(\frac{1}{8} + \frac{\sqrt{17} i}{8})}^{2}$

Step 1.2

Rewrite ${(\frac{1}{8} + \frac{\sqrt{17} i}{8})}^{2}$ as $(\frac{1}{8} + \frac{\sqrt{17} i}{8}) (\frac{1}{8} + \frac{\sqrt{17} i}{8})$ .

$(\frac{1}{8} + \frac{\sqrt{17} i}{8}) (\frac{1}{8} + \frac{\sqrt{17} i}{8})$

Step 2

Expand $(\frac{1}{8} + \frac{\sqrt{17} i}{8}) (\frac{1}{8} + \frac{\sqrt{17} i}{8})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{1}{8} (\frac{1}{8} + \frac{\sqrt{17} i}{8}) + \frac{\sqrt{17} i}{8} (\frac{1}{8} + \frac{\sqrt{17} i}{8})$

Step 2.2

Apply the distributive property.

$\frac{1}{8} \cdot \frac{1}{8} + \frac{1}{8} \cdot \frac{\sqrt{17} i}{8} + \frac{\sqrt{17} i}{8} (\frac{1}{8} + \frac{\sqrt{17} i}{8})$

Step 2.3

Apply the distributive property.

$\frac{1}{8} \cdot \frac{1}{8} + \frac{1}{8} \cdot \frac{\sqrt{17} i}{8} + \frac{\sqrt{17} i}{8} \cdot \frac{1}{8} + \frac{\sqrt{17} i}{8} \cdot \frac{\sqrt{17} i}{8}$

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 $\frac{1}{8} \cdot \frac{1}{8}$ .

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Step 3.1.1.1

Multiply $\frac{1}{8}$ by $\frac{1}{8}$ .

$\frac{1}{8 \cdot 8} + \frac{1}{8} \cdot \frac{\sqrt{17} i}{8} + \frac{\sqrt{17} i}{8} \cdot \frac{1}{8} + \frac{\sqrt{17} i}{8} \cdot \frac{\sqrt{17} i}{8}$

Step 3.1.1.2

Multiply $8$ by $8$ .

$\frac{1}{64} + \frac{1}{8} \cdot \frac{\sqrt{17} i}{8} + \frac{\sqrt{17} i}{8} \cdot \frac{1}{8} + \frac{\sqrt{17} i}{8} \cdot \frac{\sqrt{17} i}{8}$

Step 3.1.2

Multiply $\frac{1}{8} \cdot \frac{\sqrt{17} i}{8}$ .

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Step 3.1.2.1

Multiply $\frac{1}{8}$ by $\frac{\sqrt{17} i}{8}$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{8 \cdot 8} + \frac{\sqrt{17} i}{8} \cdot \frac{1}{8} + \frac{\sqrt{17} i}{8} \cdot \frac{\sqrt{17} i}{8}$

Step 3.1.2.2

Multiply $8$ by $8$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{8} \cdot \frac{1}{8} + \frac{\sqrt{17} i}{8} \cdot \frac{\sqrt{17} i}{8}$

Step 3.1.3

Multiply $\frac{\sqrt{17} i}{8} \cdot \frac{1}{8}$ .

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Step 3.1.3.1

Multiply $\frac{\sqrt{17} i}{8}$ by $\frac{1}{8}$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{8 \cdot 8} + \frac{\sqrt{17} i}{8} \cdot \frac{\sqrt{17} i}{8}$

Step 3.1.3.2

Multiply $8$ by $8$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{8} \cdot \frac{\sqrt{17} i}{8}$

Step 3.1.4

Multiply $\frac{\sqrt{17} i}{8} \cdot \frac{\sqrt{17} i}{8}$ .

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Step 3.1.4.1

Multiply $\frac{\sqrt{17} i}{8}$ by $\frac{\sqrt{17} i}{8}$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i (\sqrt{17} i)}{8 \cdot 8}$

Step 3.1.4.2

Raise $\sqrt{17}$ to the power of $1$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{1} \sqrt{17} i i}{8 \cdot 8}$

Step 3.1.4.3

Raise $\sqrt{17}$ to the power of $1$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{1} {\sqrt{17}}^{1} i i}{8 \cdot 8}$

Step 3.1.4.4

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

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{1 + 1} i i}{8 \cdot 8}$

Step 3.1.4.5

Add $1$ and $1$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{2} i i}{8 \cdot 8}$

Step 3.1.4.6

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

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{2} (i^{1} i)}{8 \cdot 8}$

Step 3.1.4.7

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

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{2} (i^{1} i^{1})}{8 \cdot 8}$

Step 3.1.4.8

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

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{2} i^{1 + 1}}{8 \cdot 8}$

Step 3.1.4.9

Add $1$ and $1$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{2} i^{2}}{8 \cdot 8}$

Step 3.1.4.10

Multiply $8$ by $8$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{\sqrt{17}}^{2} i^{2}}{64}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 3.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{{(17^{\frac{1}{2}})}^{2} i^{2}}{64}$

Step 3.1.5.1.2

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

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{17^{\frac{1}{2} \cdot 2} i^{2}}{64}$

Step 3.1.5.1.3

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

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{17^{\frac{2}{2}} i^{2}}{64}$

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.

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{17^{\frac{2}{2}} i^{2}}{64}$

Step 3.1.5.1.4.2

Rewrite the expression.

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{17^{1} i^{2}}{64}$

Step 3.1.5.1.5

Evaluate the exponent.

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{17 i^{2}}{64}$

Step 3.1.5.2

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

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{17 \cdot - 1}{64}$

Step 3.1.6

Multiply $17$ by $- 1$ .

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} + \frac{- 17}{64}$

Step 3.1.7

Move the negative in front of the fraction.

$\frac{1}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64} - \frac{17}{64}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{1 - 17}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64}$

Step 3.3

Subtract $17$ from $1$ .

$\frac{- 16}{64} + \frac{\sqrt{17} i}{64} + \frac{\sqrt{17} i}{64}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- 16}{64} + \frac{2 \sqrt{17} i}{64}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $- 16$ and $64$ .

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Step 4.1.1

Factor $16$ out of $- 16$ .

$\frac{16 (- 1)}{64} + \frac{2 \sqrt{17} i}{64}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $16$ out of $64$ .

$\frac{16 \cdot - 1}{16 \cdot 4} + \frac{2 \sqrt{17} i}{64}$

Step 4.1.2.2

Cancel the common factor.

$\frac{16 \cdot - 1}{16 \cdot 4} + \frac{2 \sqrt{17} i}{64}$

Step 4.1.2.3

Rewrite the expression.

$\frac{- 1}{4} + \frac{2 \sqrt{17} i}{64}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{1}{4} + \frac{2 \sqrt{17} i}{64}$

Step 4.3

Cancel the common factor of $2$ and $64$ .

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Step 4.3.1

Factor $2$ out of $2 \sqrt{17} i$ .

$- \frac{1}{4} + \frac{2 (\sqrt{17} i)}{64}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $64$ .

$- \frac{1}{4} + \frac{2 (\sqrt{17} i)}{2 \cdot 32}$

Step 4.3.2.2

Cancel the common factor.

$- \frac{1}{4} + \frac{2 (\sqrt{17} i)}{2 \cdot 32}$

Step 4.3.2.3

Rewrite the expression.

$- \frac{1}{4} + \frac{\sqrt{17} i}{32}$