Precalculus Examples

$z^{5} = - 243 i$

Step 1

Substitute $u$ for $z^{5}$ .

$u^{5} = - 243 i$

Step 2

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 3

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 4

Substitute the actual values of $a = 0$ and $b = - 243$ .

$| z | = \sqrt{{(- 243)}^{2}}$

Step 5

Find $| z |$ .

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Step 5.1

Raise $- 243$ to the power of $2$ .

$| z | = \sqrt{59049}$

Step 5.2

Rewrite $59049$ as $243^{2}$ .

$| z | = \sqrt{243^{2}}$

Step 5.3

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 243$

Step 6

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

$θ = \arctan (\frac{- 243}{0})$

Step 7

Since the argument is undefined and $b$ is negative, the angle of the point on the complex plane is $\frac{3 π}{2}$ .

$θ = \frac{3 π}{2}$

Step 8

Substitute the values of $θ = \frac{3 π}{2}$ and $| z | = 243$ .

$243 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 9

Replace the right side of the equation with the trigonometric form.

$u^{5} = 243 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 10

Use De Moivre's Theorem to find an equation for $u$ .

$r^{5} (c o s (5 θ) + i s i n (5 θ)) = - 243 i = 243 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 11

Equate the modulus of the trigonometric form to $r^{5}$ to find the value of $r$ .

$r^{5} = 243$

Step 12

Solve the equation for $r$ .

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Step 12.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \sqrt[5]{243}$

Step 12.2

Simplify $\sqrt[5]{243}$ .

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Step 12.2.1

Rewrite $243$ as $3^{5}$ .

$r = \sqrt[5]{3^{5}}$

Step 12.2.2

Pull terms out from under the radical, assuming real numbers.

$r = 3$

Step 13

Find the approximate value of $r$ .

$r = 3$

Step 14

Find the possible values of $θ$ .

$c o s (5 θ) = c o s (\frac{3 π}{2} + 2 π n)$ and $s i n (5 θ) = s i n (\frac{3 π}{2} + 2 π n)$

Step 15

Finding all the possible values of $θ$ leads to the equation $5 θ = \frac{3 π}{2} + 2 π n$ .

$5 θ = \frac{3 π}{2} + 2 π n$

Step 16

Find the value of $θ$ for $r = 0$ .

$5 θ = \frac{3 π}{2} + 2 π (0)$

Step 17

Solve the equation for $θ$ .

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Step 17.1

Simplify.

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Step 17.1.1

Multiply $2 π (0)$ .

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Step 17.1.1.1

Multiply $0$ by $2$ .

$5 θ = \frac{3 π}{2} + 0 π$

Step 17.1.1.2

Multiply $0$ by $π$ .

$5 θ = \frac{3 π}{2} + 0$

Step 17.1.2

Add $\frac{3 π}{2}$ and $0$ .

$5 θ = \frac{3 π}{2}$

Step 17.2

Divide each term in $5 θ = \frac{3 π}{2}$ by $5$ and simplify.

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Step 17.2.1

Divide each term in $5 θ = \frac{3 π}{2}$ by $5$ .

$\frac{5 θ}{5} = \frac{\frac{3 π}{2}}{5}$

Step 17.2.2

Simplify the left side.

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Step 17.2.2.1

Cancel the common factor of $5$ .

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Step 17.2.2.1.1

Cancel the common factor.

$\frac{5 θ}{5} = \frac{\frac{3 π}{2}}{5}$

Step 17.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{3 π}{2}}{5}$

Step 17.2.3

Simplify the right side.

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Step 17.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{3 π}{2} \cdot \frac{1}{5}$

Step 17.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{5}$ .

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Step 17.2.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{5}$ .

$θ = \frac{3 π}{2 \cdot 5}$

Step 17.2.3.2.2

Multiply $2$ by $5$ .

$θ = \frac{3 π}{10}$

Step 18

Use the values of $θ$ and $r$ to find a solution to the equation $u^{5} = - 243 i$ .

$u_{0} = 3 (\cos (\frac{3 π}{10}) + i \sin (\frac{3 π}{10}))$

Step 19

Convert the solution to rectangular form.

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Step 19.1

Simplify each term.

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Step 19.1.1

Evaluate $\cos (\frac{3 π}{10})$ .

$u_{0} = 3 (0.58778525 + i \sin (\frac{3 π}{10}))$

Step 19.1.2

Evaluate $\sin (\frac{3 π}{10})$ .

$u_{0} = 3 (0.58778525 + i \cdot 0.80901699)$

Step 19.1.3

Move $0.80901699$ to the left of $i$ .

$u_{0} = 3 (0.58778525 + 0.80901699 i)$

Step 19.2

Simplify by multiplying through.

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Step 19.2.1

Apply the distributive property.

$u_{0} = 3 \cdot 0.58778525 + 3 (0.80901699 i)$

Step 19.2.2

Multiply.

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Step 19.2.2.1

Multiply $3$ by $0.58778525$ .

$u_{0} = 1.76335575 + 3 (0.80901699 i)$

Step 19.2.2.2

Multiply $0.80901699$ by $3$ .

$u_{0} = 1.76335575 + 2.42705098 i$

Step 20

Substitute $z^{5}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{0} = 0 + 1.76335575 + 2.42705098 i$

Step 21

Find the value of $θ$ for $r = 1$ .

$5 θ = \frac{3 π}{2} + 2 π (1)$

Step 22

Solve the equation for $θ$ .

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Step 22.1

Simplify.

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Step 22.1.1

Multiply $2$ by $1$ .

$5 θ = \frac{3 π}{2} + 2 π$

Step 22.1.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 θ = \frac{3 π}{2} + 2 π \cdot \frac{2}{2}$

Step 22.1.3

Combine $2 π$ and $\frac{2}{2}$ .

$5 θ = \frac{3 π}{2} + \frac{2 π \cdot 2}{2}$

Step 22.1.4

Combine the numerators over the common denominator.

$5 θ = \frac{3 π + 2 π \cdot 2}{2}$

Step 22.1.5

Multiply $2$ by $2$ .

$5 θ = \frac{3 π + 4 π}{2}$

Step 22.1.6

Add $3 π$ and $4 π$ .

$5 θ = \frac{7 π}{2}$

Step 22.2

Divide each term in $5 θ = \frac{7 π}{2}$ by $5$ and simplify.

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Step 22.2.1

Divide each term in $5 θ = \frac{7 π}{2}$ by $5$ .

$\frac{5 θ}{5} = \frac{\frac{7 π}{2}}{5}$

Step 22.2.2

Simplify the left side.

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Step 22.2.2.1

Cancel the common factor of $5$ .

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Step 22.2.2.1.1

Cancel the common factor.

$\frac{5 θ}{5} = \frac{\frac{7 π}{2}}{5}$

Step 22.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{7 π}{2}}{5}$

Step 22.2.3

Simplify the right side.

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Step 22.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{7 π}{2} \cdot \frac{1}{5}$

Step 22.2.3.2

Multiply $\frac{7 π}{2} \cdot \frac{1}{5}$ .

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Step 22.2.3.2.1

Multiply $\frac{7 π}{2}$ by $\frac{1}{5}$ .

$θ = \frac{7 π}{2 \cdot 5}$

Step 22.2.3.2.2

Multiply $2$ by $5$ .

$θ = \frac{7 π}{10}$

Step 23

Use the values of $θ$ and $r$ to find a solution to the equation $u^{5} = - 243 i$ .

$u_{1} = 3 (\cos (\frac{7 π}{10}) + i \sin (\frac{7 π}{10}))$

Step 24

Convert the solution to rectangular form.

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Step 24.1

Simplify each term.

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Step 24.1.1

Evaluate $\cos (\frac{7 π}{10})$ .

$u_{1} = 3 (- 0.58778525 + i \sin (\frac{7 π}{10}))$

Step 24.1.2

Evaluate $\sin (\frac{7 π}{10})$ .

$u_{1} = 3 (- 0.58778525 + i \cdot 0.80901699)$

Step 24.1.3

Move $0.80901699$ to the left of $i$ .

$u_{1} = 3 (- 0.58778525 + 0.80901699 i)$

Step 24.2

Simplify by multiplying through.

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Step 24.2.1

Apply the distributive property.

$u_{1} = 3 \cdot - 0.58778525 + 3 (0.80901699 i)$

Step 24.2.2

Multiply.

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Step 24.2.2.1

Multiply $3$ by $- 0.58778525$ .

$u_{1} = - 1.76335575 + 3 (0.80901699 i)$

Step 24.2.2.2

Multiply $0.80901699$ by $3$ .

$u_{1} = - 1.76335575 + 2.42705098 i$

Step 25

Substitute $z^{5}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{1} = 0 - 1.76335575 + 2.42705098 i$

Step 26

Find the value of $θ$ for $r = 2$ .

$5 θ = \frac{3 π}{2} + 2 π (2)$

Step 27

Solve the equation for $θ$ .

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Step 27.1

Simplify.

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Step 27.1.1

Multiply $2$ by $2$ .

$5 θ = \frac{3 π}{2} + 4 π$

Step 27.1.2

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 θ = \frac{3 π}{2} + 4 π \cdot \frac{2}{2}$

Step 27.1.3

Combine $4 π$ and $\frac{2}{2}$ .

$5 θ = \frac{3 π}{2} + \frac{4 π \cdot 2}{2}$

Step 27.1.4

Combine the numerators over the common denominator.

$5 θ = \frac{3 π + 4 π \cdot 2}{2}$

Step 27.1.5

Multiply $2$ by $4$ .

$5 θ = \frac{3 π + 8 π}{2}$

Step 27.1.6

Add $3 π$ and $8 π$ .

$5 θ = \frac{11 π}{2}$

Step 27.2

Divide each term in $5 θ = \frac{11 π}{2}$ by $5$ and simplify.

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Step 27.2.1

Divide each term in $5 θ = \frac{11 π}{2}$ by $5$ .

$\frac{5 θ}{5} = \frac{\frac{11 π}{2}}{5}$

Step 27.2.2

Simplify the left side.

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Step 27.2.2.1

Cancel the common factor of $5$ .

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Step 27.2.2.1.1

Cancel the common factor.

$\frac{5 θ}{5} = \frac{\frac{11 π}{2}}{5}$

Step 27.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{11 π}{2}}{5}$

Step 27.2.3

Simplify the right side.

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Step 27.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{11 π}{2} \cdot \frac{1}{5}$

Step 27.2.3.2

Multiply $\frac{11 π}{2} \cdot \frac{1}{5}$ .

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Step 27.2.3.2.1

Multiply $\frac{11 π}{2}$ by $\frac{1}{5}$ .

$θ = \frac{11 π}{2 \cdot 5}$

Step 27.2.3.2.2

Multiply $2$ by $5$ .

$θ = \frac{11 π}{10}$

Step 28

Use the values of $θ$ and $r$ to find a solution to the equation $u^{5} = - 243 i$ .

$u_{2} = 3 (\cos (\frac{11 π}{10}) + i \sin (\frac{11 π}{10}))$

Step 29

Convert the solution to rectangular form.

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Step 29.1

Simplify each term.

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Step 29.1.1

Evaluate $\cos (\frac{11 π}{10})$ .

$u_{2} = 3 (- 0.95105651 + i \sin (\frac{11 π}{10}))$

Step 29.1.2

Evaluate $\sin (\frac{11 π}{10})$ .

$u_{2} = 3 (- 0.95105651 + i \cdot - 0.30901699)$

Step 29.1.3

Move $- 0.30901699$ to the left of $i$ .

$u_{2} = 3 (- 0.95105651 - 0.30901699 i)$

Step 29.2

Simplify by multiplying through.

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Step 29.2.1

Apply the distributive property.

$u_{2} = 3 \cdot - 0.95105651 + 3 (- 0.30901699 i)$

Step 29.2.2

Multiply.

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Step 29.2.2.1

Multiply $3$ by $- 0.95105651$ .

$u_{2} = - 2.85316954 + 3 (- 0.30901699 i)$

Step 29.2.2.2

Multiply $- 0.30901699$ by $3$ .

$u_{2} = - 2.85316954 - 0.92705098 i$

Step 30

Substitute $z^{5}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{2} = 0 - 2.85316954 - 0.92705098 i$

Step 31

Find the value of $θ$ for $r = 3$ .

$5 θ = \frac{3 π}{2} + 2 π (3)$

Step 32

Solve the equation for $θ$ .

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Step 32.1

Simplify.

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Step 32.1.1

Multiply $3$ by $2$ .

$5 θ = \frac{3 π}{2} + 6 π$

Step 32.1.2

To write $6 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 θ = \frac{3 π}{2} + 6 π \cdot \frac{2}{2}$

Step 32.1.3

Combine $6 π$ and $\frac{2}{2}$ .

$5 θ = \frac{3 π}{2} + \frac{6 π \cdot 2}{2}$

Step 32.1.4

Combine the numerators over the common denominator.

$5 θ = \frac{3 π + 6 π \cdot 2}{2}$

Step 32.1.5

Multiply $2$ by $6$ .

$5 θ = \frac{3 π + 12 π}{2}$

Step 32.1.6

Add $3 π$ and $12 π$ .

$5 θ = \frac{15 π}{2}$

Step 32.2

Divide each term in $5 θ = \frac{15 π}{2}$ by $5$ and simplify.

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Step 32.2.1

Divide each term in $5 θ = \frac{15 π}{2}$ by $5$ .

$\frac{5 θ}{5} = \frac{\frac{15 π}{2}}{5}$

Step 32.2.2

Simplify the left side.

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Step 32.2.2.1

Cancel the common factor of $5$ .

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Step 32.2.2.1.1

Cancel the common factor.

$\frac{5 θ}{5} = \frac{\frac{15 π}{2}}{5}$

Step 32.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{15 π}{2}}{5}$

Step 32.2.3

Simplify the right side.

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Step 32.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{15 π}{2} \cdot \frac{1}{5}$

Step 32.2.3.2

Cancel the common factor of $5$ .

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Step 32.2.3.2.1

Factor $5$ out of $15 π$ .

$θ = \frac{5 (3 π)}{2} \cdot \frac{1}{5}$

Step 32.2.3.2.2

Cancel the common factor.

$θ = \frac{5 (3 π)}{2} \cdot \frac{1}{5}$

Step 32.2.3.2.3

Rewrite the expression.

$θ = \frac{3 π}{2}$

Step 33

Use the values of $θ$ and $r$ to find a solution to the equation $u^{5} = - 243 i$ .

$u_{3} = 3 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 34

Convert the solution to rectangular form.

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Step 34.1

Simplify each term.

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Step 34.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$u_{3} = 3 (\cos (\frac{π}{2}) + i \sin (\frac{3 π}{2}))$

Step 34.1.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$u_{3} = 3 (0 + i \sin (\frac{3 π}{2}))$

Step 34.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$u_{3} = 3 (0 + i (- \sin (\frac{π}{2})))$

Step 34.1.4

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

$u_{3} = 3 (0 + i (- 1 \cdot 1))$

Step 34.1.5

Multiply $- 1$ by $1$ .

$u_{3} = 3 (0 + i \cdot - 1)$

Step 34.1.6

Move $- 1$ to the left of $i$ .

$u_{3} = 3 (0 - 1 \cdot i)$

Step 34.1.7

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

$u_{3} = 3 (0 - i)$

Step 34.2

Simplify the expression.

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Step 34.2.1

Subtract $i$ from $0$ .

$u_{3} = 3 (- i)$

Step 34.2.2

Multiply $- 1$ by $3$ .

$u_{3} = - 3 i$

Step 35

Substitute $z^{5}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{3} = 0 - 3 i$

Step 36

Find the value of $θ$ for $r = 4$ .

$5 θ = \frac{3 π}{2} + 2 π (4)$

Step 37

Solve the equation for $θ$ .

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Step 37.1

Simplify.

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Step 37.1.1

Multiply $4$ by $2$ .

$5 θ = \frac{3 π}{2} + 8 π$

Step 37.1.2

To write $8 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 θ = \frac{3 π}{2} + 8 π \cdot \frac{2}{2}$

Step 37.1.3

Combine $8 π$ and $\frac{2}{2}$ .

$5 θ = \frac{3 π}{2} + \frac{8 π \cdot 2}{2}$

Step 37.1.4

Combine the numerators over the common denominator.

$5 θ = \frac{3 π + 8 π \cdot 2}{2}$

Step 37.1.5

Multiply $2$ by $8$ .

$5 θ = \frac{3 π + 16 π}{2}$

Step 37.1.6

Add $3 π$ and $16 π$ .

$5 θ = \frac{19 π}{2}$

Step 37.2

Divide each term in $5 θ = \frac{19 π}{2}$ by $5$ and simplify.

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Step 37.2.1

Divide each term in $5 θ = \frac{19 π}{2}$ by $5$ .

$\frac{5 θ}{5} = \frac{\frac{19 π}{2}}{5}$

Step 37.2.2

Simplify the left side.

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Step 37.2.2.1

Cancel the common factor of $5$ .

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Step 37.2.2.1.1

Cancel the common factor.

$\frac{5 θ}{5} = \frac{\frac{19 π}{2}}{5}$

Step 37.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{19 π}{2}}{5}$

Step 37.2.3

Simplify the right side.

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Step 37.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{19 π}{2} \cdot \frac{1}{5}$

Step 37.2.3.2

Multiply $\frac{19 π}{2} \cdot \frac{1}{5}$ .

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Step 37.2.3.2.1

Multiply $\frac{19 π}{2}$ by $\frac{1}{5}$ .

$θ = \frac{19 π}{2 \cdot 5}$

Step 37.2.3.2.2

Multiply $2$ by $5$ .

$θ = \frac{19 π}{10}$

Step 38

Use the values of $θ$ and $r$ to find a solution to the equation $u^{5} = - 243 i$ .

$u_{4} = 3 (\cos (\frac{19 π}{10}) + i \sin (\frac{19 π}{10}))$

Step 39

Convert the solution to rectangular form.

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Step 39.1

Simplify each term.

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Step 39.1.1

Evaluate $\cos (\frac{19 π}{10})$ .

$u_{4} = 3 (0.95105651 + i \sin (\frac{19 π}{10}))$

Step 39.1.2

Evaluate $\sin (\frac{19 π}{10})$ .

$u_{4} = 3 (0.95105651 + i \cdot - 0.30901699)$

Step 39.1.3

Move $- 0.30901699$ to the left of $i$ .

$u_{4} = 3 (0.95105651 - 0.30901699 i)$

Step 39.2

Simplify by multiplying through.

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Step 39.2.1

Apply the distributive property.

$u_{4} = 3 \cdot 0.95105651 + 3 (- 0.30901699 i)$

Step 39.2.2

Multiply.

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Step 39.2.2.1

Multiply $3$ by $0.95105651$ .

$u_{4} = 2.85316954 + 3 (- 0.30901699 i)$

Step 39.2.2.2

Multiply $- 0.30901699$ by $3$ .

$u_{4} = 2.85316954 - 0.92705098 i$

Step 40

Substitute $z^{5}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{4} = 0 + 2.85316954 - 0.92705098 i$

Step 41

These are the complex solutions to $u^{5} = - 243 i$ .

$z_{0} = 1.76335575 + 2.42705098 i$

$z_{1} = - 1.76335575 + 2.42705098 i$

$z_{2} = - 2.85316954 - 0.92705098 i$

$z_{3} = - 3 i$

$z_{4} = 2.85316954 - 0.92705098 i$