Precalculus Examples

${(3 (\cos (27 °) + i \sin (27 °)))}^{5}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Evaluate $\cos (27 °)$ .

${(3 (0.89100652 + i \sin (27 °)))}^{5}$

Step 1.1.2

Evaluate $\sin (27 °)$ .

${(3 (0.89100652 + i \cdot 0.45399049))}^{5}$

Step 1.1.3

Move $0.45399049$ to the left of $i$ .

${(3 (0.89100652 + 0.45399049 i))}^{5}$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

${(3 \cdot 0.89100652 + 3 (0.45399049 i))}^{5}$

Step 1.2.2

Multiply.

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Step 1.2.2.1

Multiply $3$ by $0.89100652$ .

${(2.67301957 + 3 (0.45399049 i))}^{5}$

Step 1.2.2.2

Multiply $0.45399049$ by $3$ .

${(2.67301957 + 1.36197149 i)}^{5}$

Step 2

Use the Binomial Theorem.

${2.67301957}^{5} + 5 \cdot {2.67301957}^{4} (1.36197149 i) + 10 \cdot {2.67301957}^{3} {(1.36197149 i)}^{2} + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Raise $2.67301957$ to the power of $5$ .

$136.46167381 + 5 \cdot {2.67301957}^{4} (1.36197149 i) + 10 \cdot {2.67301957}^{3} {(1.36197149 i)}^{2} + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.2

Raise $2.67301957$ to the power of $4$ .

$136.46167381 + 5 \cdot 51.05150564 (1.36197149 i) + 10 \cdot {2.67301957}^{3} {(1.36197149 i)}^{2} + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.3

Multiply $5$ by $51.05150564$ .

$136.46167381 + 255.25752824 (1.36197149 i) + 10 \cdot {2.67301957}^{3} {(1.36197149 i)}^{2} + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.4

Multiply $1.36197149$ by $255.25752824$ .

$136.46167381 + 347.65347843 i + 10 \cdot {2.67301957}^{3} {(1.36197149 i)}^{2} + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.5

Raise $2.67301957$ to the power of $3$ .

$136.46167381 + 347.65347843 i + 10 \cdot 19.09881475 {(1.36197149 i)}^{2} + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.6

Multiply $10$ by $19.09881475$ .

$136.46167381 + 347.65347843 i + 190.98814753 {(1.36197149 i)}^{2} + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.7

Apply the product rule to $1.36197149 i$ .

$136.46167381 + 347.65347843 i + 190.98814753 ({1.36197149}^{2} i^{2}) + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.8

Raise $1.36197149$ to the power of $2$ .

$136.46167381 + 347.65347843 i + 190.98814753 (1.85496636 i^{2}) + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.9

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

$136.46167381 + 347.65347843 i + 190.98814753 (1.85496636 \cdot - 1) + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.10

Multiply $190.98814753 (1.85496636 \cdot - 1)$ .

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Step 3.1.10.1

Multiply $1.85496636$ by $- 1$ .

$136.46167381 + 347.65347843 i + 190.98814753 \cdot - 1.85496636 + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.10.2

Multiply $190.98814753$ by $- 1.85496636$ .

$136.46167381 + 347.65347843 i - 354.27658973 + 10 \cdot {2.67301957}^{2} {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.11

Raise $2.67301957$ to the power of $2$ .

$136.46167381 + 347.65347843 i - 354.27658973 + 10 \cdot 7.14503363 {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.12

Multiply $10$ by $7.14503363$ .

$136.46167381 + 347.65347843 i - 354.27658973 + 71.45033635 {(1.36197149 i)}^{3} + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.13

Apply the product rule to $1.36197149 i$ .

$136.46167381 + 347.65347843 i - 354.27658973 + 71.45033635 ({1.36197149}^{3} i^{3}) + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.14

Raise $1.36197149$ to the power of $3$ .

$136.46167381 + 347.65347843 i - 354.27658973 + 71.45033635 (2.52641132 i^{3}) + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.15

Factor out $i^{2}$ .

$136.46167381 + 347.65347843 i - 354.27658973 + 71.45033635 (2.52641132 (i^{2} \cdot i)) + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.16

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

$136.46167381 + 347.65347843 i - 354.27658973 + 71.45033635 (2.52641132 (- 1 \cdot i)) + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.17

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

$136.46167381 + 347.65347843 i - 354.27658973 + 71.45033635 (2.52641132 (- i)) + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.18

Multiply $- 1$ by $2.52641132$ .

$136.46167381 + 347.65347843 i - 354.27658973 + 71.45033635 (- 2.52641132 i) + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.19

Multiply $- 2.52641132$ by $71.45033635$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 5 \cdot 2.67301957 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.20

Multiply $5$ by $2.67301957$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 13.36509786 {(1.36197149 i)}^{4} + {(1.36197149 i)}^{5}$

Step 3.1.21

Apply the product rule to $1.36197149 i$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 13.36509786 ({1.36197149}^{4} i^{4}) + {(1.36197149 i)}^{5}$

Step 3.1.22

Raise $1.36197149$ to the power of $4$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 13.36509786 (3.44090021 i^{4}) + {(1.36197149 i)}^{5}$

Step 3.1.23

Rewrite $i^{4}$ as $1$ .

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Step 3.1.23.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 13.36509786 (3.44090021 {(i^{2})}^{2}) + {(1.36197149 i)}^{5}$

Step 3.1.23.2

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

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 13.36509786 (3.44090021 {(- 1)}^{2}) + {(1.36197149 i)}^{5}$

Step 3.1.23.3

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

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 13.36509786 (3.44090021 \cdot 1) + {(1.36197149 i)}^{5}$

Step 3.1.24

Multiply $13.36509786 (3.44090021 \cdot 1)$ .

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Step 3.1.24.1

Multiply $3.44090021$ by $1$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 13.36509786 \cdot 3.44090021 + {(1.36197149 i)}^{5}$

Step 3.1.24.2

Multiply $13.36509786$ by $3.44090021$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 45.98796809 + {(1.36197149 i)}^{5}$

Step 3.1.25

Apply the product rule to $1.36197149 i$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 45.98796809 + {1.36197149}^{5} i^{5}$

Step 3.1.26

Raise $1.36197149$ to the power of $5$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 45.98796809 + 4.68640802 i^{5}$

Step 3.1.27

Factor out $i^{4}$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 45.98796809 + 4.68640802 (i^{4} i)$

Step 3.1.28

Rewrite $i^{4}$ as $1$ .

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Step 3.1.28.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 45.98796809 + 4.68640802 ({(i^{2})}^{2} i)$

Step 3.1.28.2

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

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 45.98796809 + 4.68640802 ({(- 1)}^{2} i)$

Step 3.1.28.3

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

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 45.98796809 + 4.68640802 (1 i)$

Step 3.1.29

Multiply $i$ by $1$ .

$136.46167381 + 347.65347843 i - 354.27658973 - 180.51293863 i + 45.98796809 + 4.68640802 i$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Subtract $354.27658973$ from $136.46167381$ .

$- 217.81491592 + 347.65347843 i - 180.51293863 i + 45.98796809 + 4.68640802 i$

Step 3.2.2

Add $- 217.81491592$ and $45.98796809$ .

$- 171.82694782 + 347.65347843 i - 180.51293863 i + 4.68640802 i$

Step 3.2.3

Subtract $180.51293863 i$ from $347.65347843 i$ .

$- 171.82694782 + 167.1405398 i + 4.68640802 i$

Step 3.2.4

Add $167.1405398 i$ and $4.68640802 i$ .

$- 171.82694782 + 171.82694782 i$

Step 4

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 5

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 6

Substitute the actual values of $a = - 171.82694782$ and $b = 171.82694782$ .

$| z | = \sqrt{{171.82694782}^{2} + {(- 171.82694782)}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

Raise $171.82694782$ to the power of $2$ .

$| z | = \sqrt{29524.4\overline{9} + {(- 171.82694782)}^{2}}$

Step 7.2

Raise $- 171.82694782$ to the power of $2$ .

$| z | = \sqrt{29524.4\overline{9} + 29524.4\overline{9}}$

Step 7.3

Add $29524.4\overline{9}$ and $29524.4\overline{9}$ .

$| z | = \sqrt{59048.\overline{9}}$

Step 8

Evaluate the root.

$| z | = 242.\overline{9}$

Step 9

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{171.82694782}{- 171.82694782})$

Step 10

The inverse tangent of $\frac{171.82694782}{- 171.82694782}$ is $- 0.78539816$ .

$θ = - 0.78539816$

Step 11

Substitute the values of $θ = - 0.78539816$ and $| z | = 242.\overline{9}$ .

$242.\overline{9} (\cos (- 0.78539816) + i \sin (- 0.78539816))$