Calculus Examples

${(5 (\cos (30 °) + i \sin (30 °)))}^{3}$

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 $\cos (30 °)$ is $\frac{\sqrt{3}}{2}$ .

${(5 (\frac{\sqrt{3}}{2} + i \sin (30 °)))}^{3}$

Step 1.1.2

The exact value of $\sin (30 °)$ is $\frac{1}{2}$ .

${(5 (\frac{\sqrt{3}}{2} + i \frac{1}{2}))}^{3}$

Step 1.1.3

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

${(5 (\frac{\sqrt{3}}{2} + \frac{i}{2}))}^{3}$

Step 1.2

Simplify terms.

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Step 1.2.1

Apply the distributive property.

${(5 \frac{\sqrt{3}}{2} + 5 \frac{i}{2})}^{3}$

Step 1.2.2

Combine $5$ and $\frac{\sqrt{3}}{2}$ .

${(\frac{5 \sqrt{3}}{2} + 5 \frac{i}{2})}^{3}$

Step 1.2.3

Combine $5$ and $\frac{i}{2}$ .

${(\frac{5 \sqrt{3}}{2} + \frac{5 i}{2})}^{3}$

Step 2

Use the Binomial Theorem.

${(\frac{5 \sqrt{3}}{2})}^{3} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Apply the product rule to $\frac{5 \sqrt{3}}{2}$ .

$\frac{{(5 \sqrt{3})}^{3}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.1.2

Apply the product rule to $5 \sqrt{3}$ .

$\frac{5^{3} {\sqrt{3}}^{3}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.2

Simplify the numerator.

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Step 3.1.2.1

Raise $5$ to the power of $3$ .

$\frac{125 {\sqrt{3}}^{3}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.2.2

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

$\frac{125 \sqrt{3^{3}}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.2.3

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

$\frac{125 \sqrt{27}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.2.4

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

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Step 3.1.2.4.1

Factor $9$ out of $27$ .

$\frac{125 \sqrt{9 (3)}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.2.4.2

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

$\frac{125 \sqrt{3^{2} \cdot 3}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.2.5

Pull terms out from under the radical.

$\frac{125 \cdot 3 \sqrt{3}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.2.6

Multiply $125$ by $3$ .

$\frac{375 \sqrt{3}}{2^{3}} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.3

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

$\frac{375 \sqrt{3}}{8} + 3 {(\frac{5 \sqrt{3}}{2})}^{2} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.4

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

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Step 3.1.4.1

Apply the product rule to $\frac{5 \sqrt{3}}{2}$ .

$\frac{375 \sqrt{3}}{8} + 3 \frac{{(5 \sqrt{3})}^{2}}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.4.2

Apply the product rule to $5 \sqrt{3}$ .

$\frac{375 \sqrt{3}}{8} + 3 \frac{5^{2} {\sqrt{3}}^{2}}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Raise $5$ to the power of $2$ .

$\frac{375 \sqrt{3}}{8} + 3 \frac{25 {\sqrt{3}}^{2}}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.5.2

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

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Step 3.1.5.2.1

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

$\frac{375 \sqrt{3}}{8} + 3 \frac{25 {(3^{\frac{1}{2}})}^{2}}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.5.2.2

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

$\frac{375 \sqrt{3}}{8} + 3 \frac{25 \cdot 3^{\frac{1}{2} \cdot 2}}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.5.2.3

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

$\frac{375 \sqrt{3}}{8} + 3 \frac{25 \cdot 3^{\frac{2}{2}}}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.5.2.4

Cancel the common factor of $2$ .

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Step 3.1.5.2.4.1

Cancel the common factor.

$\frac{375 \sqrt{3}}{8} + 3 \frac{25 \cdot 3^{\frac{2}{2}}}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.5.2.4.2

Rewrite the expression.

$\frac{375 \sqrt{3}}{8} + 3 \frac{25 \cdot 3^{1}}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.5.2.5

Evaluate the exponent.

$\frac{375 \sqrt{3}}{8} + 3 \frac{25 \cdot 3}{2^{2}} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.6

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

$\frac{375 \sqrt{3}}{8} + 3 \frac{25 \cdot 3}{4} \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.7

Multiply $25$ by $3$ .

$\frac{375 \sqrt{3}}{8} + 3 (\frac{75}{4}) \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.8

Multiply $3 (\frac{75}{4})$ .

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Step 3.1.8.1

Combine $3$ and $\frac{75}{4}$ .

$\frac{375 \sqrt{3}}{8} + \frac{3 \cdot 75}{4} \cdot \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.8.2

Multiply $3$ by $75$ .

$\frac{375 \sqrt{3}}{8} + \frac{225}{4} \cdot \frac{5 i}{2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.9

Multiply $\frac{225}{4} \cdot \frac{5 i}{2}$ .

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Step 3.1.9.1

Multiply $\frac{225}{4}$ by $\frac{5 i}{2}$ .

$\frac{375 \sqrt{3}}{8} + \frac{225 (5 i)}{4 \cdot 2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.9.2

Multiply $5$ by $225$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{4 \cdot 2} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.9.3

Multiply $4$ by $2$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + 3 \frac{5 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.10

Multiply $3 \frac{5 \sqrt{3}}{2}$ .

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Step 3.1.10.1

Combine $3$ and $\frac{5 \sqrt{3}}{2}$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{3 (5 \sqrt{3})}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.10.2

Multiply $5$ by $3$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3}}{2} {(\frac{5 i}{2})}^{2} + {(\frac{5 i}{2})}^{3}$

Step 3.1.11

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

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Step 3.1.11.1

Apply the product rule to $\frac{5 i}{2}$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3}}{2} \cdot \frac{{(5 i)}^{2}}{2^{2}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.11.2

Apply the product rule to $5 i$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3}}{2} \cdot \frac{5^{2} i^{2}}{2^{2}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.12

Combine.

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3} (5^{2} i^{2})}{2 \cdot 2^{2}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.13

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

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Step 3.1.13.1

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

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Step 3.1.13.1.1

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

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3} (5^{2} i^{2})}{2^{1} \cdot 2^{2}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.13.1.2

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

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3} (5^{2} i^{2})}{2^{1 + 2}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.13.2

Add $1$ and $2$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3} (5^{2} i^{2})}{2^{3}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.14

Simplify the numerator.

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Step 3.1.14.1

Raise $5$ to the power of $2$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3} \cdot 25 i^{2}}{2^{3}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.14.2

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

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{15 \sqrt{3} \cdot 25 \cdot - 1}{2^{3}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.14.3

Combine exponents.

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Step 3.1.14.3.1

Multiply $25$ by $15$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{375 \sqrt{3} \cdot - 1}{2^{3}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.14.3.2

Multiply $- 1$ by $375$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{- 375 \sqrt{3}}{2^{3}} + {(\frac{5 i}{2})}^{3}$

Step 3.1.15

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

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} + \frac{- 375 \sqrt{3}}{8} + {(\frac{5 i}{2})}^{3}$

Step 3.1.16

Move the negative in front of the fraction.

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + {(\frac{5 i}{2})}^{3}$

Step 3.1.17

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

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Step 3.1.17.1

Apply the product rule to $\frac{5 i}{2}$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{{(5 i)}^{3}}{2^{3}}$

Step 3.1.17.2

Apply the product rule to $5 i$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{5^{3} i^{3}}{2^{3}}$

Step 3.1.18

Simplify the numerator.

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Step 3.1.18.1

Raise $5$ to the power of $3$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{125 i^{3}}{2^{3}}$

Step 3.1.18.2

Factor out $i^{2}$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{125 (i^{2} \cdot i)}{2^{3}}$

Step 3.1.18.3

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

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{125 (- 1 \cdot i)}{2^{3}}$

Step 3.1.18.4

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

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{125 \cdot - 1 i}{2^{3}}$

Step 3.1.18.5

Combine exponents.

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Step 3.1.18.5.1

Factor out negative.

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{- (125 i)}{2^{3}}$

Step 3.1.18.5.2

Multiply $125$ by $- 1$ .

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{- 125 i}{2^{3}}$

Step 3.1.19

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

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} + \frac{- 125 i}{8}$

Step 3.1.20

Move the negative in front of the fraction.

$\frac{375 \sqrt{3}}{8} + \frac{1125 i}{8} - \frac{375 \sqrt{3}}{8} - \frac{125 i}{8}$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

$\frac{375 \sqrt{3} + 1125 i - 375 \sqrt{3} - 125 i}{8}$

Step 3.2.2

Subtract $375 \sqrt{3}$ from $375 \sqrt{3}$ .

$\frac{0 + 1125 i - 125 i}{8}$

Step 3.2.3

Add $0$ and $1125 i$ .

$\frac{1125 i - 125 i}{8}$

Step 3.2.4

Subtract $125 i$ from $1125 i$ .

$\frac{1000 i}{8}$

Step 3.2.5

Cancel the common factor of $1000$ and $8$ .

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Step 3.2.5.1

Factor $8$ out of $1000 i$ .

$\frac{8 (125 i)}{8}$

Step 3.2.5.2

Cancel the common factors.

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Step 3.2.5.2.1

Factor $8$ out of $8$ .

$\frac{8 (125 i)}{8 (1)}$

Step 3.2.5.2.2

Cancel the common factor.

$\frac{8 (125 i)}{8 \cdot 1}$

Step 3.2.5.2.3

Rewrite the expression.

$\frac{125 i}{1}$

Step 3.2.5.2.4

Divide $125 i$ by $1$ .

$125 i$