Precalculus Examples

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

Step 1

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

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

Step 2

Use the Binomial Theorem.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 3.1.2

One to any power is one.

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

Step 3.1.3

Raise $2$ to the power of $3$ .

$\frac{1}{8} + 3 {(\frac{1}{2})}^{2} \frac{\sqrt{3} i}{2} + 3 (\frac{1}{2}) {(\frac{\sqrt{3} i}{2})}^{2} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.4

Apply the product rule to $\frac{1}{2}$ .

$\frac{1}{8} + 3 \frac{1^{2}}{2^{2}} \frac{\sqrt{3} i}{2} + 3 (\frac{1}{2}) {(\frac{\sqrt{3} i}{2})}^{2} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.5

One to any power is one.

$\frac{1}{8} + 3 \frac{1}{2^{2}} \frac{\sqrt{3} i}{2} + 3 (\frac{1}{2}) {(\frac{\sqrt{3} i}{2})}^{2} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.6

Raise $2$ to the power of $2$ .

$\frac{1}{8} + 3 (\frac{1}{4}) \frac{\sqrt{3} i}{2} + 3 (\frac{1}{2}) {(\frac{\sqrt{3} i}{2})}^{2} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.7

Combine $3$ and $\frac{1}{4}$ .

$\frac{1}{8} + \frac{3}{4} \cdot \frac{\sqrt{3} i}{2} + 3 (\frac{1}{2}) {(\frac{\sqrt{3} i}{2})}^{2} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.8

Multiply $\frac{3}{4} \cdot \frac{\sqrt{3} i}{2}$ .

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Step 3.1.8.1

Multiply $\frac{3}{4}$ by $\frac{\sqrt{3} i}{2}$ .

$\frac{1}{8} + \frac{3 (\sqrt{3} i)}{4 \cdot 2} + 3 (\frac{1}{2}) {(\frac{\sqrt{3} i}{2})}^{2} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.8.2

Multiply $4$ by $2$ .

$\frac{1}{8} + \frac{3 (\sqrt{3} i)}{8} + 3 (\frac{1}{2}) {(\frac{\sqrt{3} i}{2})}^{2} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.9

Combine $3$ and $\frac{1}{2}$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3}{2} {(\frac{\sqrt{3} i}{2})}^{2} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.10.1

Apply the product rule to $\frac{\sqrt{3} i}{2}$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3}{2} \cdot \frac{{(\sqrt{3} i)}^{2}}{2^{2}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.10.2

Apply the product rule to $\sqrt{3} i$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3}{2} \cdot \frac{{\sqrt{3}}^{2} i^{2}}{2^{2}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.11

Combine.

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 ({\sqrt{3}}^{2} i^{2})}{2 \cdot 2^{2}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.12

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 3.1.12.1

Multiply $2$ by $2^{2}$ .

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Step 3.1.12.1.1

Raise $2$ to the power of $1$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 ({\sqrt{3}}^{2} i^{2})}{2^{1} \cdot 2^{2}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.12.1.2

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

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 ({\sqrt{3}}^{2} i^{2})}{2^{1 + 2}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.12.2

Add $1$ and $2$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 ({\sqrt{3}}^{2} i^{2})}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13

Simplify the numerator.

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Step 3.1.13.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.13.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 {(3^{\frac{1}{2}})}^{2} i^{2}}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13.1.2

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

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 \cdot 3^{\frac{1}{2} \cdot 2} i^{2}}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13.1.3

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

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 \cdot 3^{\frac{2}{2}} i^{2}}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13.1.4

Cancel the common factor of $2$ .

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Step 3.1.13.1.4.1

Cancel the common factor.

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 \cdot 3^{\frac{2}{2}} i^{2}}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13.1.4.2

Rewrite the expression.

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 \cdot 3^{1} i^{2}}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13.1.5

Evaluate the exponent.

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 \cdot 3 i^{2}}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13.2

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

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{3 \cdot 3 \cdot - 1}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13.3

Combine exponents.

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Step 3.1.13.3.1

Multiply $3$ by $3$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{9 \cdot - 1}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.13.3.2

Multiply $9$ by $- 1$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{- 9}{2^{3}} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.14

Raise $2$ to the power of $3$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} + \frac{- 9}{8} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.15

Move the negative in front of the fraction.

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + {(\frac{\sqrt{3} i}{2})}^{3}$

Step 3.1.16

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.16.1

Apply the product rule to $\frac{\sqrt{3} i}{2}$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{{(\sqrt{3} i)}^{3}}{2^{3}}$

Step 3.1.16.2

Apply the product rule to $\sqrt{3} i$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{{\sqrt{3}}^{3} i^{3}}{2^{3}}$

Step 3.1.17

Simplify the numerator.

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Step 3.1.17.1

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{\sqrt{3^{3}} i^{3}}{2^{3}}$

Step 3.1.17.2

Raise $3$ to the power of $3$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{\sqrt{27} i^{3}}{2^{3}}$

Step 3.1.17.3

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.1.17.3.1

Factor $9$ out of $27$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{\sqrt{9 (3)} i^{3}}{2^{3}}$

Step 3.1.17.3.2

Rewrite $9$ as $3^{2}$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{\sqrt{3^{2} \cdot 3} i^{3}}{2^{3}}$

Step 3.1.17.4

Pull terms out from under the radical.

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{3 \sqrt{3} i^{3}}{2^{3}}$

Step 3.1.17.5

Factor out $i^{2}$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{3 \sqrt{3} (i^{2} \cdot i)}{2^{3}}$

Step 3.1.17.6

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

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{3 \sqrt{3} (- 1 \cdot i)}{2^{3}}$

Step 3.1.17.7

Rewrite $- 1 i$ as $- i$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{3 \sqrt{3} \cdot - 1 i}{2^{3}}$

Step 3.1.17.8

Combine exponents.

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Step 3.1.17.8.1

Factor out negative.

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{- (3 \sqrt{3} i)}{2^{3}}$

Step 3.1.17.8.2

Multiply $3$ by $- 1$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{- 3 (\sqrt{3} i)}{2^{3}}$

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{- 3 \sqrt{3} i}{2^{3}}$

Step 3.1.18

Raise $2$ to the power of $3$ .

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} + \frac{- 3 \sqrt{3} i}{8}$

Step 3.1.19

Move the negative in front of the fraction.

$\frac{1}{8} + \frac{3 \sqrt{3} i}{8} - \frac{9}{8} - \frac{3 \sqrt{3} i}{8}$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

$\frac{1 + 3 \sqrt{3} i - 9 - 3 \sqrt{3} i}{8}$

Step 3.2.2

Subtract $9$ from $1$ .

$\frac{3 \sqrt{3} i - 8 - 3 \sqrt{3} i}{8}$

Step 3.2.3

Subtract $3 \sqrt{3} i$ from $3 \sqrt{3} i$ .

$\frac{0 - 8}{8}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Subtract $8$ from $0$ .

$\frac{- 8}{8}$

Step 3.2.4.2

Divide $- 8$ by $8$ .

$- 1$