Examples

{(z - 3)}^{4} = 2 i

${(z - 3)}^{4} = 2 i$

Step 1

Substitute

u

$u$ for

z - 3

$z - 3$ .

u^{4} = 2 i

$u^{4} = 2 i$

Step 2

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

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

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

Step 3

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

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

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

z = a + b i

$z = a + b i$

Step 4

Substitute the actual values of

a = 0

$a = 0$ and

b = 2

$b = 2$ .

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

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

Step 5

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

| z | = 2

$| z | = 2$

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{2}{0})

$θ = \arctan (\frac{2}{0})$

Step 7

Since the argument is undefined and

b

$b$ is positive, the angle of the point on the complex plane is

\frac{π}{2}

$\frac{π}{2}$ .

θ = \frac{π}{2}

$θ = \frac{π}{2}$

Step 8

Substitute the values of

θ = \frac{π}{2}

$θ = \frac{π}{2}$ and

| z | = 2

$| z | = 2$ .

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

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

Step 9

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

u^{4} = 2 (cos (\frac{π}{2}) + i sin (\frac{π}{2}))

$u^{4} = 2 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 10

Use De Moivre's Theorem to find an equation for

u

$u$ .

r^{4} (c o s (4 θ) + i s i n (4 θ)) = 2 i = 2 (cos (\frac{π}{2}) + i sin (\frac{π}{2}))

$r^{4} (c o s (4 θ) + i s i n (4 θ)) = 2 i = 2 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 11

Equate the modulus of the trigonometric form to

r^{4}

$r^{4}$ to find the value of

r

$r$ .

r^{4} = 2

$r^{4} = 2$

Step 12

Solve the equation for

r

$r$ .

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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 = \pm \sqrt[4]{2}

$r = \pm \sqrt[4]{2}$

Step 12.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 12.2.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

r = \sqrt[4]{2}

$r = \sqrt[4]{2}$

Step 12.2.2

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

r = - \sqrt[4]{2}

$r = - \sqrt[4]{2}$

Step 12.2.3

The complete solution is the result of both the positive and negative portions of the solution.

r = \sqrt[4]{2}, - \sqrt[4]{2}

$r = \sqrt[4]{2}, - \sqrt[4]{2}$

r = \sqrt[4]{2}, - \sqrt[4]{2}

$r = \sqrt[4]{2}, - \sqrt[4]{2}$

r = \sqrt[4]{2}, - \sqrt[4]{2}

$r = \sqrt[4]{2}, - \sqrt[4]{2}$

Step 13

Find the approximate value of

r

$r$ .

r = 1.18920711

$r = 1.18920711$

Step 14

Find the possible values of

θ

$θ$ .

c o s (4 θ) = c o s (\frac{π}{2} + 2 π n)

$c o s (4 θ) = c o s (\frac{π}{2} + 2 π n)$ and

s i n (4 θ) = s i n (\frac{π}{2} + 2 π n)

$s i n (4 θ) = s i n (\frac{π}{2} + 2 π n)$

Step 15

Finding all the possible values of

θ

$θ$ leads to the equation

4 θ = \frac{π}{2} + 2 π n

$4 θ = \frac{π}{2} + 2 π n$ .

4 θ = \frac{π}{2} + 2 π n

$4 θ = \frac{π}{2} + 2 π n$

Step 16

Find the value of

θ

$θ$ for

r = 0

$r = 0$ .

4 θ = \frac{π}{2} + 2 π (0)

$4 θ = \frac{π}{2} + 2 π (0)$

Step 17

Solve the equation for

θ

$θ$ .

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Step 17.1

Simplify.

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Step 17.1.1

Multiply

2 π (0)

$2 π (0)$ .

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Step 17.1.1.1

Multiply

0

$0$ by

2

$2$ .

4 θ = \frac{π}{2} + 0 π

$4 θ = \frac{π}{2} + 0 π$

Step 17.1.1.2

Multiply

0

$0$ by

π

$π$ .

4 θ = \frac{π}{2} + 0

$4 θ = \frac{π}{2} + 0$

4 θ = \frac{π}{2} + 0

$4 θ = \frac{π}{2} + 0$

Step 17.1.2

Add

\frac{π}{2}

$\frac{π}{2}$ and

0

$0$ .

4 θ = \frac{π}{2}

$4 θ = \frac{π}{2}$

4 θ = \frac{π}{2}

$4 θ = \frac{π}{2}$

Step 17.2

Divide each term in

4 θ = \frac{π}{2}

$4 θ = \frac{π}{2}$ by

4

$4$ and simplify.

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Step 17.2.1

Divide each term in

4 θ = \frac{π}{2}

$4 θ = \frac{π}{2}$ by

4

$4$ .

\frac{4 θ}{4} = \frac{\frac{π}{2}}{4}

$\frac{4 θ}{4} = \frac{\frac{π}{2}}{4}$

Step 17.2.2

Simplify the left side.

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Step 17.2.2.1

Cancel the common factor of

4

$4$ .

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Step 17.2.2.1.1

Cancel the common factor.

$\frac{4 θ}{4} = \frac{\frac{π}{2}}{4}$

Step 17.2.2.1.2

Divide

$θ$ by

$1$ .

$θ = \frac{\frac{π}{2}}{4}$

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.

$θ = \frac{π}{2} \cdot \frac{1}{4}$

Step 17.2.3.2

Multiply

$\frac{π}{2} \cdot \frac{1}{4}$ .

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Step 17.2.3.2.1

Multiply

$\frac{π}{2}$ by

$\frac{1}{4}$ .

$θ = \frac{π}{2 \cdot 4}$

Step 17.2.3.2.2

Multiply

$2$ by

$4$ .

$θ = \frac{π}{8}$

Step 18

Use the values of

$θ$ and

$r$ to find a solution to the equation

$u^{4} = 2 i$ .

$u_{0} = 1.18920711 (\cos (\frac{π}{8}) + i \sin (\frac{π}{8}))$

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

The exact value of

$\cos (\frac{π}{8})$ is

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

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Step 19.1.1.1

Rewrite

$\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$u_{0} = 1.18920711 (\cos (\frac{\frac{π}{4}}{2}) + i \sin (\frac{π}{8}))$

Step 19.1.1.2

Apply the cosine half-angle identity

$\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$u_{0} = 1.18920711 (\pm \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.3

Change the

$\pm$ to

$+$ because cosine is positive in the first quadrant.

$u_{0} = 1.18920711 (\sqrt{\frac{1 + \cos (\frac{π}{4})}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.4

The exact value of

$\cos (\frac{π}{4})$ is

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

$u_{0} = 1.18920711 (\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5

Simplify

$\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ .

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Step 19.1.1.5.1

Write

$1$ as a fraction with a common denominator.

$u_{0} = 1.18920711 (\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.2

Combine the numerators over the common denominator.

$u_{0} = 1.18920711 (\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.3

Multiply the numerator by the reciprocal of the denominator.

$u_{0} = 1.18920711 (\sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.4

Multiply

$\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 19.1.1.5.4.1

Multiply

$\frac{2 + \sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$u_{0} = 1.18920711 (\sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.4.2

Multiply

$2$ by

$2$ .

$u_{0} = 1.18920711 (\sqrt{\frac{2 + \sqrt{2}}{4}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.5

Rewrite

$\sqrt{\frac{2 + \sqrt{2}}{4}}$ as

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.6

Simplify the denominator.

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Step 19.1.1.5.6.1

Rewrite

$4$ as

$2^{2}$ .

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.6.2

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

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sin (\frac{π}{8}))$

Step 19.1.2

The exact value of

$\sin (\frac{π}{8})$ is

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

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Step 19.1.2.1

Rewrite

$\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sin (\frac{\frac{π}{4}}{2}))$

Step 19.1.2.2

Apply the sine half-angle identity.

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i (\pm \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}))$

Step 19.1.2.3

Change the

$\pm$ to

$+$ because sine is positive in the first quadrant.

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}})$

Step 19.1.2.4

Simplify

$\sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}$ .

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Step 19.1.2.4.1

The exact value of

$\cos (\frac{π}{4})$ is

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

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}})$

Step 19.1.2.4.2

Write

$1$ as a fraction with a common denominator.

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}})$

Step 19.1.2.4.3

Combine the numerators over the common denominator.

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}})$

Step 19.1.2.4.4

Multiply the numerator by the reciprocal of the denominator.

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}})$

Step 19.1.2.4.5

Multiply

$\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 19.1.2.4.5.1

Multiply

$\frac{2 - \sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}})$

Step 19.1.2.4.5.2

Multiply

$2$ by

$2$ .

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{2 - \sqrt{2}}{4}})$

Step 19.1.2.4.6

Rewrite

$\sqrt{\frac{2 - \sqrt{2}}{4}}$ as

$\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i (\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}))$

Step 19.1.2.4.7

Simplify the denominator.

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Step 19.1.2.4.7.1

Rewrite

$4$ as

$2^{2}$ .

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i (\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}}))$

Step 19.1.2.4.7.2

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

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i (\frac{\sqrt{2 - \sqrt{2}}}{2}))$

Step 19.1.3

Combine

$i$ and

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

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{i \sqrt{2 - \sqrt{2}}}{2})$

Step 19.2

Simplify terms.

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Step 19.2.1

Combine the numerators over the common denominator.

$u_{0} = 1.18920711 (\frac{\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}}}{2})$

Step 19.2.2

Combine

$1.18920711$ and

$\frac{\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}}}{2}$ .

$u_{0} = \frac{1.18920711 (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})}{2}$

Step 19.2.3

Factor

$2$ out of

$2$ .

$u_{0} = \frac{1.18920711 (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})}{2 (1)}$

Step 19.3

Separate fractions.

$u_{0} = \frac{1.18920711}{2} \cdot \frac{\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}}}{1}$

Step 19.4

Simplify the expression.

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Step 19.4.1

Divide

$1.18920711$ by

$2$ .

$u_{0} = 0.59460355 (\frac{\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}}}{1})$

Step 19.4.2

Divide

$\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}}$ by

$1$ .

$u_{0} = 0.59460355 (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})$

Step 19.5

Apply the distributive property.

$u_{0} = 0.59460355 \sqrt{2 + \sqrt{2}} + 0.59460355 (i \sqrt{2 - \sqrt{2}})$

Step 19.6

Multiply

$0.59460355$ by

$\sqrt{2 + \sqrt{2}}$ .

$u_{0} = 1.09868411 + 0.59460355 (i \sqrt{2 - \sqrt{2}})$

Step 19.7

Multiply

$\sqrt{2 - \sqrt{2}}$ by

$0.59460355$ .

$u_{0} = 1.09868411 + 0.45508986 i$

Step 20

Substitute

$z - 3$ for

$u$ to calculate the value of

$z$ after the right shift.

$z_{0} = 3 + 1.09868411 + 0.45508986 i$

Step 21

Find the value of

$θ$ for

$r = 1$ .

$4 θ = \frac{π}{2} + 2 π (1)$

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

$1$ .

$4 θ = \frac{π}{2} + 2 π$

Step 22.1.2

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 22.1.3

Combine

$2 π$ and

$\frac{2}{2}$ .

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

Step 22.1.4

Combine the numerators over the common denominator.

$4 θ = \frac{π + 2 π \cdot 2}{2}$

Step 22.1.5

Multiply

$2$ by

$2$ .

$4 θ = \frac{π + 4 π}{2}$

Step 22.1.6

Add

$π$ and

$4 π$ .

$4 θ = \frac{5 π}{2}$

Step 22.2

Divide each term in

$4 θ = \frac{5 π}{2}$ by

$4$ and simplify.

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Step 22.2.1

Divide each term in

$4 θ = \frac{5 π}{2}$ by

$4$ .

$\frac{4 θ}{4} = \frac{\frac{5 π}{2}}{4}$

Step 22.2.2

Simplify the left side.

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Step 22.2.2.1

Cancel the common factor of

$4$ .

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Step 22.2.2.1.1

Cancel the common factor.

$\frac{4 θ}{4} = \frac{\frac{5 π}{2}}{4}$

Step 22.2.2.1.2

Divide

$θ$ by

$1$ .

$θ = \frac{\frac{5 π}{2}}{4}$

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.

$θ = \frac{5 π}{2} \cdot \frac{1}{4}$

Step 22.2.3.2

Multiply

$\frac{5 π}{2} \cdot \frac{1}{4}$ .

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Step 22.2.3.2.1

Multiply

$\frac{5 π}{2}$ by

$\frac{1}{4}$ .

$θ = \frac{5 π}{2 \cdot 4}$

Step 22.2.3.2.2

Multiply

$2$ by

$4$ .

$θ = \frac{5 π}{8}$

Step 23

Use the values of

$θ$ and

$r$ to find a solution to the equation

$u^{4} = 2 i$ .

$u_{1} = 1.18920711 (\cos (\frac{5 π}{8}) + i \sin (\frac{5 π}{8}))$

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

The exact value of

$\cos (\frac{5 π}{8})$ is

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

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Step 24.1.1.1

Rewrite

$\frac{5 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$u_{1} = 1.18920711 (\cos (\frac{\frac{5 π}{4}}{2}) + i \sin (\frac{5 π}{8}))$

Step 24.1.1.2

Apply the cosine half-angle identity

$\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$u_{1} = 1.18920711 (\pm \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.3

Change the

$\pm$ to

$-$ because cosine is negative in the second quadrant.

$u_{1} = 1.18920711 (- \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4

Simplify

$- \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}}$ .

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Step 24.1.1.4.1

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

$u_{1} = 1.18920711 (- \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.2

The exact value of

$\cos (\frac{π}{4})$ is

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

$u_{1} = 1.18920711 (- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.3

Write

$1$ as a fraction with a common denominator.

$u_{1} = 1.18920711 (- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.4

Combine the numerators over the common denominator.

$u_{1} = 1.18920711 (- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

$u_{1} = 1.18920711 (- \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.6

Multiply

$\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 24.1.1.4.6.1

Multiply

$\frac{2 - \sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$u_{1} = 1.18920711 (- \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.6.2

Multiply

$2$ by

$2$ .

$u_{1} = 1.18920711 (- \sqrt{\frac{2 - \sqrt{2}}{4}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.7

Rewrite

$\sqrt{\frac{2 - \sqrt{2}}{4}}$ as

$\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.8

Simplify the denominator.

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Step 24.1.1.4.8.1

Rewrite

$4$ as

$2^{2}$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.8.2

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

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sin (\frac{5 π}{8}))$

Step 24.1.2

The exact value of

$\sin (\frac{5 π}{8})$ is

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

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Step 24.1.2.1

Rewrite

$\frac{5 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sin (\frac{\frac{5 π}{4}}{2}))$

Step 24.1.2.2

Apply the sine half-angle identity.

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i (\pm \sqrt{\frac{1 - \cos (\frac{5 π}{4})}{2}}))$

Step 24.1.2.3

Change the

$\pm$ to

$+$ because sine is positive in the second quadrant.

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{1 - \cos (\frac{5 π}{4})}{2}})$

Step 24.1.2.4

Simplify

$\sqrt{\frac{1 - \cos (\frac{5 π}{4})}{2}}$ .

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Step 24.1.2.4.1

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

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}})$

Step 24.1.2.4.2

The exact value of

$\cos (\frac{π}{4})$ is

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

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}})$

Step 24.1.2.4.3

Multiply

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

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Step 24.1.2.4.3.1

Multiply

$- 1$ by

$- 1$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{1 + 1 (\frac{\sqrt{2}}{2})}{2}})$

Step 24.1.2.4.3.2

Multiply

$\frac{\sqrt{2}}{2}$ by

$1$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}})$

Step 24.1.2.4.4

Write

$1$ as a fraction with a common denominator.

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}})$

Step 24.1.2.4.5

Combine the numerators over the common denominator.

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}})$

Step 24.1.2.4.6

Multiply the numerator by the reciprocal of the denominator.

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}})$

Step 24.1.2.4.7

Multiply

$\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 24.1.2.4.7.1

Multiply

$\frac{2 + \sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}})$

Step 24.1.2.4.7.2

Multiply

$2$ by

$2$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sqrt{\frac{2 + \sqrt{2}}{4}})$

Step 24.1.2.4.8

Rewrite

$\sqrt{\frac{2 + \sqrt{2}}{4}}$ as

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i (\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}))$

Step 24.1.2.4.9

Simplify the denominator.

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Step 24.1.2.4.9.1

Rewrite

$4$ as

$2^{2}$ .

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i (\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}}))$

Step 24.1.2.4.9.2

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

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i (\frac{\sqrt{2 + \sqrt{2}}}{2}))$

Step 24.1.3

Combine

$i$ and

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

$u_{1} = 1.18920711 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + \frac{i \sqrt{2 + \sqrt{2}}}{2})$

Step 24.2

Simplify terms.

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Step 24.2.1

Combine the numerators over the common denominator.

$u_{1} = 1.18920711 (\frac{- \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}}}{2})$

Step 24.2.2

Combine

$1.18920711$ and

$\frac{- \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}}}{2}$ .

$u_{1} = \frac{1.18920711 (- \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}})}{2}$

Step 24.2.3

Factor

$2$ out of

$2$ .

$u_{1} = \frac{1.18920711 (- \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}})}{2 (1)}$

Step 24.3

Separate fractions.

$u_{1} = \frac{1.18920711}{2} \cdot \frac{- \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}}}{1}$

Step 24.4

Simplify the expression.

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Step 24.4.1

Divide

$1.18920711$ by

$2$ .

$u_{1} = 0.59460355 (\frac{- \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}}}{1})$

Step 24.4.2

Divide

$- \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}}$ by

$1$ .

$u_{1} = 0.59460355 (- \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}})$

Step 24.5

Apply the distributive property.

$u_{1} = 0.59460355 (- \sqrt{2 - \sqrt{2}}) + 0.59460355 (i \sqrt{2 + \sqrt{2}})$

Step 24.6

Multiply

$0.59460355 (- \sqrt{2 - \sqrt{2}})$ .

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Step 24.6.1

Multiply

$- 1$ by

$0.59460355$ .

$u_{1} = - 0.59460355 \sqrt{2 - \sqrt{2}} + 0.59460355 (i \sqrt{2 + \sqrt{2}})$

Step 24.6.2

Multiply

$- 0.59460355$ by

$\sqrt{2 - \sqrt{2}}$ .

$u_{1} = - 0.45508986 + 0.59460355 (i \sqrt{2 + \sqrt{2}})$

Step 24.7

Multiply

$\sqrt{2 + \sqrt{2}}$ by

$0.59460355$ .

$u_{1} = - 0.45508986 + 1.09868411 i$

Step 25

Substitute

$z - 3$ for

$u$ to calculate the value of

$z$ after the right shift.

$z_{1} = 3 - 0.45508986 + 1.09868411 i$

Step 26

Find the value of

$θ$ for

$r = 2$ .

$4 θ = \frac{π}{2} + 2 π (2)$

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

$2$ .

$4 θ = \frac{π}{2} + 4 π$

Step 27.1.2

To write

$4 π$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 27.1.3

Combine

$4 π$ and

$\frac{2}{2}$ .

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

Step 27.1.4

Combine the numerators over the common denominator.

$4 θ = \frac{π + 4 π \cdot 2}{2}$

Step 27.1.5

Multiply

$2$ by

$4$ .

$4 θ = \frac{π + 8 π}{2}$

Step 27.1.6

Add

$π$ and

$8 π$ .

$4 θ = \frac{9 π}{2}$

Step 27.2

Divide each term in

$4 θ = \frac{9 π}{2}$ by

$4$ and simplify.

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Step 27.2.1

Divide each term in

$4 θ = \frac{9 π}{2}$ by

$4$ .

$\frac{4 θ}{4} = \frac{\frac{9 π}{2}}{4}$

Step 27.2.2

Simplify the left side.

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Step 27.2.2.1

Cancel the common factor of

$4$ .

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Step 27.2.2.1.1

Cancel the common factor.

$\frac{4 θ}{4} = \frac{\frac{9 π}{2}}{4}$

Step 27.2.2.1.2

Divide

$θ$ by

$1$ .

$θ = \frac{\frac{9 π}{2}}{4}$

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.

$θ = \frac{9 π}{2} \cdot \frac{1}{4}$

Step 27.2.3.2

Multiply

$\frac{9 π}{2} \cdot \frac{1}{4}$ .

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Step 27.2.3.2.1

Multiply

$\frac{9 π}{2}$ by

$\frac{1}{4}$ .

$θ = \frac{9 π}{2 \cdot 4}$

Step 27.2.3.2.2

Multiply

$2$ by

$4$ .

$θ = \frac{9 π}{8}$

Step 28

Use the values of

$θ$ and

$r$ to find a solution to the equation

$u^{4} = 2 i$ .

$u_{2} = 1.18920711 (\cos (\frac{9 π}{8}) + i \sin (\frac{9 π}{8}))$

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

The exact value of

$\cos (\frac{9 π}{8})$ is

$- \frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 29.1.1.1

Rewrite

$\frac{9 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$u_{2} = 1.18920711 (\cos (\frac{\frac{9 π}{4}}{2}) + i \sin (\frac{9 π}{8}))$

Step 29.1.1.2

Apply the cosine half-angle identity

$\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$u_{2} = 1.18920711 (\pm \sqrt{\frac{1 + \cos (\frac{9 π}{4})}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.3

Change the

$\pm$ to

$-$ because cosine is negative in the third quadrant.

$u_{2} = 1.18920711 (- \sqrt{\frac{1 + \cos (\frac{9 π}{4})}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4

Simplify

$- \sqrt{\frac{1 + \cos (\frac{9 π}{4})}{2}}$ .

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Step 29.1.1.4.1

Subtract full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$u_{2} = 1.18920711 (- \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.2

The exact value of

$\cos (\frac{π}{4})$ is

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

$u_{2} = 1.18920711 (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.3

Write

$1$ as a fraction with a common denominator.

$u_{2} = 1.18920711 (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.4

Combine the numerators over the common denominator.

$u_{2} = 1.18920711 (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

$u_{2} = 1.18920711 (- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.6

Multiply

$\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 29.1.1.4.6.1

Multiply

$\frac{2 + \sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$u_{2} = 1.18920711 (- \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.6.2

Multiply

$2$ by

$2$ .

$u_{2} = 1.18920711 (- \sqrt{\frac{2 + \sqrt{2}}{4}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.7

Rewrite

$\sqrt{\frac{2 + \sqrt{2}}{4}}$ as

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.8

Simplify the denominator.

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Step 29.1.1.4.8.1

Rewrite

$4$ as

$2^{2}$ .

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.8.2

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

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i \sin (\frac{9 π}{8}))$

Step 29.1.2

The exact value of

$\sin (\frac{9 π}{8})$ is

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

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Step 29.1.2.1

Rewrite

$\frac{9 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i \sin (\frac{\frac{9 π}{4}}{2}))$

Step 29.1.2.2

Apply the sine half-angle identity.

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (\pm \sqrt{\frac{1 - \cos (\frac{9 π}{4})}{2}}))$

Step 29.1.2.3

Change the

$\pm$ to

$-$ because sine is negative in the third quadrant.

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{1 - \cos (\frac{9 π}{4})}{2}}))$

Step 29.1.2.4

Simplify

$- \sqrt{\frac{1 - \cos (\frac{9 π}{4})}{2}}$ .

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Step 29.1.2.4.1

Subtract full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}))$

Step 29.1.2.4.2

The exact value of

$\cos (\frac{π}{4})$ is

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

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}))$

Step 29.1.2.4.3

Write

$1$ as a fraction with a common denominator.

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}}))$

Step 29.1.2.4.4

Combine the numerators over the common denominator.

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}))$

Step 29.1.2.4.5

Multiply the numerator by the reciprocal of the denominator.

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}}))$

Step 29.1.2.4.6

Multiply

$\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 29.1.2.4.6.1

Multiply

$\frac{2 - \sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}}))$

Step 29.1.2.4.6.2

Multiply

$2$ by

$2$ .

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{2 - \sqrt{2}}{4}}))$

Step 29.1.2.4.7

Rewrite

$\sqrt{\frac{2 - \sqrt{2}}{4}}$ as

$\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}))$

Step 29.1.2.4.8

Simplify the denominator.

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Step 29.1.2.4.8.1

Rewrite

$4$ as

$2^{2}$ .

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}}))$

Step 29.1.2.4.8.2

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

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \frac{\sqrt{2 - \sqrt{2}}}{2}))$

Step 29.1.3

Combine

$i$ and

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

$u_{2} = 1.18920711 (- \frac{\sqrt{2 + \sqrt{2}}}{2} - \frac{i \sqrt{2 - \sqrt{2}}}{2})$

Step 29.2

Simplify terms.

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Step 29.2.1

Combine the numerators over the common denominator.

$u_{2} = 1.18920711 (\frac{- \sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}}}{2})$

Step 29.2.2

Combine

$1.18920711$ and

$\frac{- \sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}}}{2}$ .

$u_{2} = \frac{1.18920711 (- \sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}})}{2}$

Step 29.2.3

Factor

$2$ out of

$2$ .

$u_{2} = \frac{1.18920711 (- \sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}})}{2 (1)}$

Step 29.3

Separate fractions.

$u_{2} = \frac{1.18920711}{2} \cdot \frac{- \sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}}}{1}$

Step 29.4

Simplify the expression.

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Step 29.4.1

Divide

$1.18920711$ by

$2$ .

$u_{2} = 0.59460355 (\frac{- \sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}}}{1})$

Step 29.4.2

Divide

$- \sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}}$ by

$1$ .

$u_{2} = 0.59460355 (- \sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}})$

Step 29.5

Apply the distributive property.

$u_{2} = 0.59460355 (- \sqrt{2 + \sqrt{2}}) + 0.59460355 (- i \sqrt{2 - \sqrt{2}})$

Step 29.6

Multiply

$0.59460355 (- \sqrt{2 + \sqrt{2}})$ .

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Step 29.6.1

Multiply

$- 1$ by

$0.59460355$ .

$u_{2} = - 0.59460355 \sqrt{2 + \sqrt{2}} + 0.59460355 (- i \sqrt{2 - \sqrt{2}})$

Step 29.6.2

Multiply

$- 0.59460355$ by

$\sqrt{2 + \sqrt{2}}$ .

$u_{2} = - 1.09868411 + 0.59460355 (- i \sqrt{2 - \sqrt{2}})$

Step 29.7

Multiply

$0.59460355 (- i \sqrt{2 - \sqrt{2}})$ .

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Step 29.7.1

Multiply

$- 1$ by

$0.59460355$ .

$u_{2} = - 1.09868411 - 0.59460355 (i \sqrt{2 - \sqrt{2}})$

Step 29.7.2

Multiply

$\sqrt{2 - \sqrt{2}}$ by

$- 0.59460355$ .

$u_{2} = - 1.09868411 - 0.45508986 i$

Step 30

Substitute

$z - 3$ for

$u$ to calculate the value of

$z$ after the right shift.

$z_{2} = 3 - 1.09868411 - 0.45508986 i$

Step 31

Find the value of

$θ$ for

$r = 3$ .

$4 θ = \frac{π}{2} + 2 π (3)$

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

$2$ .

$4 θ = \frac{π}{2} + 6 π$

Step 32.1.2

To write

$6 π$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$4 θ = \frac{π}{2} + 6 π \cdot \frac{2}{2}$

Step 32.1.3

Combine

$6 π$ and

$\frac{2}{2}$ .

$4 θ = \frac{π}{2} + \frac{6 π \cdot 2}{2}$

Step 32.1.4

Combine the numerators over the common denominator.

$4 θ = \frac{π + 6 π \cdot 2}{2}$

Step 32.1.5

Multiply

$2$ by

$6$ .

$4 θ = \frac{π + 12 π}{2}$

Step 32.1.6

Add

$π$ and

$12 π$ .

$4 θ = \frac{13 π}{2}$

Step 32.2

Divide each term in

$4 θ = \frac{13 π}{2}$ by

$4$ and simplify.

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Step 32.2.1

Divide each term in

$4 θ = \frac{13 π}{2}$ by

$4$ .

$\frac{4 θ}{4} = \frac{\frac{13 π}{2}}{4}$

Step 32.2.2

Simplify the left side.

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Step 32.2.2.1

Cancel the common factor of

$4$ .

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Step 32.2.2.1.1

Cancel the common factor.

$\frac{4 θ}{4} = \frac{\frac{13 π}{2}}{4}$

Step 32.2.2.1.2

Divide

$θ$ by

$1$ .

$θ = \frac{\frac{13 π}{2}}{4}$

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.

$θ = \frac{13 π}{2} \cdot \frac{1}{4}$

Step 32.2.3.2

Multiply

$\frac{13 π}{2} \cdot \frac{1}{4}$ .

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Step 32.2.3.2.1

Multiply

$\frac{13 π}{2}$ by

$\frac{1}{4}$ .

$θ = \frac{13 π}{2 \cdot 4}$

Step 32.2.3.2.2

Multiply

$2$ by

$4$ .

$θ = \frac{13 π}{8}$

Step 33

Use the values of

$θ$ and

$r$ to find a solution to the equation

$u^{4} = 2 i$ .

$u_{3} = 1.18920711 (\cos (\frac{13 π}{8}) + i \sin (\frac{13 π}{8}))$

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

The exact value of

$\cos (\frac{13 π}{8})$ is

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

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Step 34.1.1.1

Rewrite

$\frac{13 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$u_{3} = 1.18920711 (\cos (\frac{\frac{13 π}{4}}{2}) + i \sin (\frac{13 π}{8}))$

Step 34.1.1.2

Apply the cosine half-angle identity

$\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$u_{3} = 1.18920711 (\pm \sqrt{\frac{1 + \cos (\frac{13 π}{4})}{2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.3

Change the

$\pm$ to

$+$ because cosine is positive in the fourth quadrant.

$u_{3} = 1.18920711 (\sqrt{\frac{1 + \cos (\frac{13 π}{4})}{2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4

Simplify

$\sqrt{\frac{1 + \cos (\frac{13 π}{4})}{2}}$ .

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Step 34.1.1.4.1

Subtract full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$u_{3} = 1.18920711 (\sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.2

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

$u_{3} = 1.18920711 (\sqrt{\frac{1 - \cos (\frac{π}{4})}{2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.3

The exact value of

$\cos (\frac{π}{4})$ is

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

$u_{3} = 1.18920711 (\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.4

Write

$1$ as a fraction with a common denominator.

$u_{3} = 1.18920711 (\sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.5

Combine the numerators over the common denominator.

$u_{3} = 1.18920711 (\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.6

Multiply the numerator by the reciprocal of the denominator.

$u_{3} = 1.18920711 (\sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.7

Multiply

$\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 34.1.1.4.7.1

Multiply

$\frac{2 - \sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$u_{3} = 1.18920711 (\sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.7.2

Multiply

$2$ by

$2$ .

$u_{3} = 1.18920711 (\sqrt{\frac{2 - \sqrt{2}}{4}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.8

Rewrite

$\sqrt{\frac{2 - \sqrt{2}}{4}}$ as

$\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.9

Simplify the denominator.

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Step 34.1.1.4.9.1

Rewrite

$4$ as

$2^{2}$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}} + i \sin (\frac{13 π}{8}))$

Step 34.1.1.4.9.2

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

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i \sin (\frac{13 π}{8}))$

Step 34.1.2

The exact value of

$\sin (\frac{13 π}{8})$ is

$- \frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 34.1.2.1

Rewrite

$\frac{13 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i \sin (\frac{\frac{13 π}{4}}{2}))$

Step 34.1.2.2

Apply the sine half-angle identity.

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (\pm \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{2}}))$

Step 34.1.2.3

Change the

$\pm$ to

$-$ because sine is negative in the fourth quadrant.

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{2}}))$

Step 34.1.2.4

Simplify

$- \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{2}}$ .

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Step 34.1.2.4.1

Subtract full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{1 - \cos (\frac{5 π}{4})}{2}}))$

Step 34.1.2.4.2

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

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}}))$

Step 34.1.2.4.3

The exact value of

$\cos (\frac{π}{4})$ is

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

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}))$

Step 34.1.2.4.4

Multiply

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

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Step 34.1.2.4.4.1

Multiply

$- 1$ by

$- 1$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{1 + 1 (\frac{\sqrt{2}}{2})}{2}}))$

Step 34.1.2.4.4.2

Multiply

$\frac{\sqrt{2}}{2}$ by

$1$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}))$

Step 34.1.2.4.5

Write

$1$ as a fraction with a common denominator.

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}}))$

Step 34.1.2.4.6

Combine the numerators over the common denominator.

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}))$

Step 34.1.2.4.7

Multiply the numerator by the reciprocal of the denominator.

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}}))$

Step 34.1.2.4.8

Multiply

$\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 34.1.2.4.8.1

Multiply

$\frac{2 + \sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}}))$

Step 34.1.2.4.8.2

Multiply

$2$ by

$2$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \sqrt{\frac{2 + \sqrt{2}}{4}}))$

Step 34.1.2.4.9

Rewrite

$\sqrt{\frac{2 + \sqrt{2}}{4}}$ as

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}))$

Step 34.1.2.4.10

Simplify the denominator.

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Step 34.1.2.4.10.1

Rewrite

$4$ as

$2^{2}$ .

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}}))$

Step 34.1.2.4.10.2

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

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} + i (- \frac{\sqrt{2 + \sqrt{2}}}{2}))$

Step 34.1.3

Combine

$i$ and

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

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}}}{2} - \frac{i \sqrt{2 + \sqrt{2}}}{2})$

Step 34.2

Simplify terms.

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Step 34.2.1

Combine the numerators over the common denominator.

$u_{3} = 1.18920711 (\frac{\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}}}{2})$

Step 34.2.2

Combine

$1.18920711$ and

$\frac{\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}}}{2}$ .

$u_{3} = \frac{1.18920711 (\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}})}{2}$

Step 34.2.3

Factor

$2$ out of

$2$ .

$u_{3} = \frac{1.18920711 (\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}})}{2 (1)}$

Step 34.3

Separate fractions.

$u_{3} = \frac{1.18920711}{2} \cdot \frac{\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}}}{1}$

Step 34.4

Simplify the expression.

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Step 34.4.1

Divide

$1.18920711$ by

$2$ .

$u_{3} = 0.59460355 (\frac{\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}}}{1})$

Step 34.4.2

Divide

$\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}}$ by

$1$ .

$u_{3} = 0.59460355 (\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}})$

Step 34.5

Apply the distributive property.

$u_{3} = 0.59460355 \sqrt{2 - \sqrt{2}} + 0.59460355 (- i \sqrt{2 + \sqrt{2}})$

Step 34.6

Multiply

$0.59460355$ by

$\sqrt{2 - \sqrt{2}}$ .

$u_{3} = 0.45508986 + 0.59460355 (- i \sqrt{2 + \sqrt{2}})$

Step 34.7

Multiply

$0.59460355 (- i \sqrt{2 + \sqrt{2}})$ .

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Step 34.7.1

Multiply

$- 1$ by

$0.59460355$ .

$u_{3} = 0.45508986 - 0.59460355 (i \sqrt{2 + \sqrt{2}})$

Step 34.7.2

Multiply

$\sqrt{2 + \sqrt{2}}$ by

$- 0.59460355$ .

$u_{3} = 0.45508986 - 1.09868411 i$

Step 35

Substitute

$z - 3$ for

$u$ to calculate the value of

$z$ after the right shift.

$z_{3} = 3 + 0.45508986 - 1.09868411 i$

Step 36

These are the complex solutions to

$u^{4} = 2 i$ .

$z_{0} = 4.09868411 + 0.45508986 i$

$z_{1} = 2.54491013 + 1.09868411 i$

$z_{2} = 1.90131588 - 0.45508986 i$

$z_{3} = 3.45508986 - 1.09868411 i$

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