Precalculus Examples

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

Step 1

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

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

Step 2

Use the Binomial Theorem.

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

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

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

Step 3.1.2

Simplify the numerator.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

$\frac{\sqrt{27}}{2^{3}} + 3 {(\frac{\sqrt{3}}{2})}^{2} \frac{i}{2} + 3 \frac{\sqrt{3}}{2} {(\frac{i}{2})}^{2} + {(\frac{i}{2})}^{3}$

Step 3.1.2.3

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

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Step 3.1.2.3.1

Factor $9$ out of $27$ .

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

Step 3.1.2.3.2

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

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

Step 3.1.2.4

Pull terms out from under the radical.

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

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

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

Step 3.1.5.3

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

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

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

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

Step 3.1.5.4.2

Rewrite the expression.

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

Step 3.1.5.5

Evaluate the exponent.

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

Step 3.1.6

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

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

Step 3.1.7

Multiply $3 (\frac{3}{4})$ .

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Step 3.1.7.1

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

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

Step 3.1.7.2

Multiply $3$ by $3$ .

$\frac{3 \sqrt{3}}{8} + \frac{9}{4} \cdot \frac{i}{2} + 3 \frac{\sqrt{3}}{2} {(\frac{i}{2})}^{2} + {(\frac{i}{2})}^{3}$

Step 3.1.8

Multiply $\frac{9}{4} \cdot \frac{i}{2}$ .

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Step 3.1.8.1

Multiply $\frac{9}{4}$ by $\frac{i}{2}$ .

$\frac{3 \sqrt{3}}{8} + \frac{9 i}{4 \cdot 2} + 3 \frac{\sqrt{3}}{2} {(\frac{i}{2})}^{2} + {(\frac{i}{2})}^{3}$

Step 3.1.8.2

Multiply $4$ by $2$ .

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

Step 3.1.9

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

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

Step 3.1.10

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

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

Step 3.1.11

Combine.

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

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 $1$ .

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

Step 3.1.12.1.2

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

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

Step 3.1.12.2

Add $1$ and $2$ .

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

Step 3.1.13

Simplify the numerator.

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Step 3.1.13.1

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

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

Step 3.1.13.2

Multiply $- 1$ by $3$ .

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

Step 3.1.14

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

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

Step 3.1.15

Move the negative in front of the fraction.

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

Step 3.1.16

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

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

Step 3.1.17

Simplify the numerator.

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Step 3.1.17.1

Factor out $i^{2}$ .

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

Step 3.1.17.2

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

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

Step 3.1.17.3

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

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

Step 3.1.18

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

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

Step 3.1.19

Move the negative in front of the fraction.

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

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

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

$\frac{0 + 9 i - i}{8}$

Step 3.2.3

Add $0$ and $9 i$ .

$\frac{9 i - i}{8}$

Step 3.2.4

Subtract $i$ from $9 i$ .

$\frac{8 i}{8}$

Step 3.2.5

Cancel the common factor of $8$ .

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Step 3.2.5.1

Cancel the common factor.

$\frac{8 i}{8}$

Step 3.2.5.2

Divide $i$ by $1$ .

$i$