Calculus Examples

${(6 (\cos (60 °) + i \sin (60 °)))}^{3}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

The exact value of $\cos (60 °)$ is $\frac{1}{2}$ .

${(6 (\frac{1}{2} + i \sin (60 °)))}^{3}$

Step 1.1.2

The exact value of $\sin (60 °)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 1.1.3

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

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

Step 1.2

Simplify terms.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Factor $2$ out of $6$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Factor $2$ out of $6$ .

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

Step 1.2.3.2

Cancel the common factor.

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

Step 1.2.3.3

Rewrite the expression.

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

Step 2

Use the Binomial Theorem.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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

Step 3.1.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 3.1.2.1

Multiply $3$ by $3^{2}$ .

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Step 3.1.2.1.1

Raise $3$ to the power of $1$ .

$27 + 3^{1} \cdot 3^{2} (3 i \sqrt{3}) + 3 \cdot 3 {(3 i \sqrt{3})}^{2} + {(3 i \sqrt{3})}^{3}$

Step 3.1.2.1.2

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

$27 + 3^{1 + 2} (3 i \sqrt{3}) + 3 \cdot 3 {(3 i \sqrt{3})}^{2} + {(3 i \sqrt{3})}^{3}$

Step 3.1.2.2

Add $1$ and $2$ .

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

Step 3.1.3

Multiply $3^{3}$ by $3$ by adding the exponents.

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Step 3.1.3.1

Move $3$ .

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

Step 3.1.3.2

Multiply $3$ by $3^{3}$ .

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Step 3.1.3.2.1

Raise $3$ to the power of $1$ .

$27 + 3^{1} \cdot 3^{3} (i \sqrt{3}) + 3 \cdot 3 {(3 i \sqrt{3})}^{2} + {(3 i \sqrt{3})}^{3}$

Step 3.1.3.2.2

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

$27 + 3^{1 + 3} (i \sqrt{3}) + 3 \cdot 3 {(3 i \sqrt{3})}^{2} + {(3 i \sqrt{3})}^{3}$

Step 3.1.3.3

Add $1$ and $3$ .

$27 + 3^{4} (i \sqrt{3}) + 3 \cdot 3 {(3 i \sqrt{3})}^{2} + {(3 i \sqrt{3})}^{3}$

Step 3.1.4

Raise $3$ to the power of $4$ .

$27 + 81 (i \sqrt{3}) + 3 \cdot 3 {(3 i \sqrt{3})}^{2} + {(3 i \sqrt{3})}^{3}$

Step 3.1.5

Multiply $3$ by $3$ .

$27 + 81 i \sqrt{3} + 9 {(3 i \sqrt{3})}^{2} + {(3 i \sqrt{3})}^{3}$

Step 3.1.6

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

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Step 3.1.6.1

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

$27 + 81 i \sqrt{3} + 9 ({(3 i)}^{2} {\sqrt{3}}^{2}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.6.2

Apply the product rule to $3 i$ .

$27 + 81 i \sqrt{3} + 9 (3^{2} i^{2} {\sqrt{3}}^{2}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.7

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

$27 + 81 i \sqrt{3} + 9 (9 i^{2} {\sqrt{3}}^{2}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.8

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

$27 + 81 i \sqrt{3} + 9 (9 \cdot - 1 {\sqrt{3}}^{2}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.9

Multiply $9$ by $- 1$ .

$27 + 81 i \sqrt{3} + 9 (- 9 {\sqrt{3}}^{2}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.10

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

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Step 3.1.10.1

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

$27 + 81 i \sqrt{3} + 9 (- 9 {(3^{\frac{1}{2}})}^{2}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.10.2

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

$27 + 81 i \sqrt{3} + 9 (- 9 \cdot 3^{\frac{1}{2} \cdot 2}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.10.3

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

$27 + 81 i \sqrt{3} + 9 (- 9 \cdot 3^{\frac{2}{2}}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.10.4

Cancel the common factor of $2$ .

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Step 3.1.10.4.1

Cancel the common factor.

$27 + 81 i \sqrt{3} + 9 (- 9 \cdot 3^{\frac{2}{2}}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.10.4.2

Rewrite the expression.

$27 + 81 i \sqrt{3} + 9 (- 9 \cdot 3^{1}) + {(3 i \sqrt{3})}^{3}$

Step 3.1.10.5

Evaluate the exponent.

$27 + 81 i \sqrt{3} + 9 (- 9 \cdot 3) + {(3 i \sqrt{3})}^{3}$

Step 3.1.11

Multiply $9 (- 9 \cdot 3)$ .

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Step 3.1.11.1

Multiply $- 9$ by $3$ .

$27 + 81 i \sqrt{3} + 9 \cdot - 27 + {(3 i \sqrt{3})}^{3}$

Step 3.1.11.2

Multiply $9$ by $- 27$ .

$27 + 81 i \sqrt{3} - 243 + {(3 i \sqrt{3})}^{3}$

Step 3.1.12

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

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Step 3.1.12.1

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

$27 + 81 i \sqrt{3} - 243 + {(3 i)}^{3} {\sqrt{3}}^{3}$

Step 3.1.12.2

Apply the product rule to $3 i$ .

$27 + 81 i \sqrt{3} - 243 + 3^{3} i^{3} {\sqrt{3}}^{3}$

Step 3.1.13

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

$27 + 81 i \sqrt{3} - 243 + 27 i^{3} {\sqrt{3}}^{3}$

Step 3.1.14

Factor out $i^{2}$ .

$27 + 81 i \sqrt{3} - 243 + 27 (i^{2} \cdot i) {\sqrt{3}}^{3}$

Step 3.1.15

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

$27 + 81 i \sqrt{3} - 243 + 27 (- 1 \cdot i) {\sqrt{3}}^{3}$

Step 3.1.16

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

$27 + 81 i \sqrt{3} - 243 + 27 (- i) {\sqrt{3}}^{3}$

Step 3.1.17

Multiply $- 1$ by $27$ .

$27 + 81 i \sqrt{3} - 243 - 27 i {\sqrt{3}}^{3}$

Step 3.1.18

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

$27 + 81 i \sqrt{3} - 243 - 27 i \sqrt{3^{3}}$

Step 3.1.19

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

$27 + 81 i \sqrt{3} - 243 - 27 i \sqrt{27}$

Step 3.1.20

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

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Step 3.1.20.1

Factor $9$ out of $27$ .

$27 + 81 i \sqrt{3} - 243 - 27 i \sqrt{9 (3)}$

Step 3.1.20.2

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

$27 + 81 i \sqrt{3} - 243 - 27 i \sqrt{3^{2} \cdot 3}$

Step 3.1.21

Pull terms out from under the radical.

$27 + 81 i \sqrt{3} - 243 - 27 i (3 \sqrt{3})$

Step 3.1.22

Multiply $3$ by $- 27$ .

$27 + 81 i \sqrt{3} - 243 - 81 i \sqrt{3}$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Subtract $243$ from $27$ .

$81 i \sqrt{3} - 216 - 81 i \sqrt{3}$

Step 3.2.2

Subtract $81 i \sqrt{3}$ from $81 i \sqrt{3}$ .

$0 - 216$

Step 3.2.3

Subtract $216$ from $0$ .

$- 216$