Precalculus Examples

$z^{4} = 81 i$

Step 1

Substitute $u$ for $z^{4}$ .

$u^{4} = 81 i$

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 $b = 81$ .

$| z | = \sqrt{81^{2}}$

Step 5

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

$| z | = 81$

Step 6

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

$θ = \arctan (\frac{81}{0})$

Step 7

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

$θ = \frac{π}{2}$

Step 8

Substitute the values of $θ = \frac{π}{2}$ and $| z | = 81$ .

$81 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 9

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

$u^{4} = 81 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 10

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

$r^{4} (c o s (4 θ) + i s i n (4 θ)) = 81 i = 81 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 11

Equate the modulus of the trigonometric form to $r^{4}$ to find the value of $r$ .

$r^{4} = 81$

Step 12

Solve the equation for $r$ .

Tap for more steps...

Step 12.1

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

$r = \pm \sqrt[4]{81}$

Step 12.2

Simplify $\pm \sqrt[4]{81}$ .

Tap for more steps...

Step 12.2.1

Rewrite $81$ as $3^{4}$ .

$r = \pm \sqrt[4]{3^{4}}$

Step 12.2.2

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

$r = \pm 3$

Step 12.3

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

Tap for more steps...

Step 12.3.1

First, use the positive value of the $\pm$ to find the first solution.

$r = 3$

Step 12.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$r = - 3$

Step 12.3.3

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

$r = 3, - 3$

Step 13

Find the approximate value of $r$ .

$r = 3$

Step 14

Find the possible values of $θ$ .

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

Step 15

Finding all the possible values of $θ$ leads to the equation $4 θ = \frac{π}{2} + 2 π n$ .

$4 θ = \frac{π}{2} + 2 π n$

Step 16

Find the value of $θ$ for $r = 0$ .

$4 θ = \frac{π}{2} + 2 π (0)$

Step 17

Solve the equation for $θ$ .

Tap for more steps...

Step 17.1

Simplify.

Tap for more steps...

Step 17.1.1

Multiply $2 π (0)$ .

Tap for more steps...

Step 17.1.1.1

Multiply $0$ by $2$ .

$4 θ = \frac{π}{2} + 0 π$

Step 17.1.1.2

Multiply $0$ by $π$ .

$4 θ = \frac{π}{2} + 0$

Step 17.1.2

Add $\frac{π}{2}$ and $0$ .

$4 θ = \frac{π}{2}$

Step 17.2

Divide each term in $4 θ = \frac{π}{2}$ by $4$ and simplify.

Tap for more steps...

Step 17.2.1

Divide each term in $4 θ = \frac{π}{2}$ by $4$ .

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

Step 17.2.2

Simplify the left side.

Tap for more steps...

Step 17.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

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.

Tap for more steps...

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 $\frac{π}{2} \cdot \frac{1}{4}$ .

Tap for more steps...

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 $u^{4} = 81 i$ .

$u_{0} = 3 (\cos (\frac{π}{8}) + i \sin (\frac{π}{8}))$

Step 19

Convert the solution to rectangular form.

Tap for more steps...

Step 19.1

Simplify each term.

Tap for more steps...

Step 19.1.1

The exact value of $\cos (\frac{π}{8})$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

Tap for more steps...

Step 19.1.1.1

Rewrite $\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$u_{0} = 3 (\cos (\frac{\frac{π}{4}}{2}) + i \sin (\frac{π}{8}))$

Step 19.1.1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$u_{0} = 3 (\pm \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

$u_{0} = 3 (\sqrt{\frac{1 + \cos (\frac{π}{4})}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.4

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$u_{0} = 3 (\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5

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

Tap for more steps...

Step 19.1.1.5.1

Write $1$ as a fraction with a common denominator.

$u_{0} = 3 (\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.2

Combine the numerators over the common denominator.

$u_{0} = 3 (\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.3

Multiply the numerator by the reciprocal of the denominator.

$u_{0} = 3 (\sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.4

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

Tap for more steps...

Step 19.1.1.5.4.1

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

$u_{0} = 3 (\sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.4.2

Multiply $2$ by $2$ .

$u_{0} = 3 (\sqrt{\frac{2 + \sqrt{2}}{4}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.5

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

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.6

Simplify the denominator.

Tap for more steps...

Step 19.1.1.5.6.1

Rewrite $4$ as $2^{2}$ .

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}} + i \sin (\frac{π}{8}))$

Step 19.1.1.5.6.2

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

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sin (\frac{π}{8}))$

Step 19.1.2

The exact value of $\sin (\frac{π}{8})$ is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

Tap for more steps...

Step 19.1.2.1

Rewrite $\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sin (\frac{\frac{π}{4}}{2}))$

Step 19.1.2.2

Apply the sine half-angle identity.

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i (\pm \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}))$

Step 19.1.2.3

Change the $\pm$ to $+$ because sine is positive in the first quadrant.

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}})$

Step 19.1.2.4

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

Tap for more steps...

Step 19.1.2.4.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}})$

Step 19.1.2.4.2

Write $1$ as a fraction with a common denominator.

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

Step 19.1.2.4.3

Combine the numerators over the common denominator.

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

Step 19.1.2.4.4

Multiply the numerator by the reciprocal of the denominator.

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}})$

Step 19.1.2.4.5

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

Tap for more steps...

Step 19.1.2.4.5.1

Multiply $\frac{2 - \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}})$

Step 19.1.2.4.5.2

Multiply $2$ by $2$ .

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i \sqrt{\frac{2 - \sqrt{2}}{4}})$

Step 19.1.2.4.6

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

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i (\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}))$

Step 19.1.2.4.7

Simplify the denominator.

Tap for more steps...

Step 19.1.2.4.7.1

Rewrite $4$ as $2^{2}$ .

$u_{0} = 3 (\frac{\sqrt{2 + \sqrt{2}}}{2} + i (\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}}))$

Step 19.1.2.4.7.2

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

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

Step 19.1.3

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

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

Step 19.2

Combine fractions.

Tap for more steps...

Step 19.2.1

Combine the numerators over the common denominator.

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

Step 19.2.2

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

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

Step 20

Substitute $z^{4}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{0} = 0 + \frac{3 (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})}{2}$

Step 21

Find the value of $θ$ for $r = 1$ .

$4 θ = \frac{π}{2} + 2 π (1)$

Step 22

Solve the equation for $θ$ .

Tap for more steps...

Step 22.1

Simplify.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 22.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

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.

Tap for more steps...

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 $\frac{5 π}{2} \cdot \frac{1}{4}$ .

Tap for more steps...

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 $u^{4} = 81 i$ .

$u_{1} = 3 (\cos (\frac{5 π}{8}) + i \sin (\frac{5 π}{8}))$

Step 24

Convert the solution to rectangular form.

Tap for more steps...

Step 24.1

Simplify each term.

Tap for more steps...

Step 24.1.1

The exact value of $\cos (\frac{5 π}{8})$ is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ .

Tap for more steps...

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 $2$ .

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

Step 24.1.1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

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

Step 24.1.1.3

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

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

Step 24.1.1.4

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

Tap for more steps...

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.

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

Step 24.1.1.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$u_{1} = 3 (- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.3

Write $1$ as a fraction with a common denominator.

$u_{1} = 3 (- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.4

Combine the numerators over the common denominator.

$u_{1} = 3 (- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

$u_{1} = 3 (- \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.6

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

Tap for more steps...

Step 24.1.1.4.6.1

Multiply $\frac{2 - \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$u_{1} = 3 (- \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.6.2

Multiply $2$ by $2$ .

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

Step 24.1.1.4.7

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

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

Step 24.1.1.4.8

Simplify the denominator.

Tap for more steps...

Step 24.1.1.4.8.1

Rewrite $4$ as $2^{2}$ .

$u_{1} = 3 (- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}} + i \sin (\frac{5 π}{8}))$

Step 24.1.1.4.8.2

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

$u_{1} = 3 (- \frac{\sqrt{2 - \sqrt{2}}}{2} + i \sin (\frac{5 π}{8}))$

Step 24.1.2

The exact value of $\sin (\frac{5 π}{8})$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

Tap for more steps...

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 $2$ .

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

Step 24.1.2.2

Apply the sine half-angle identity.

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

Step 24.1.2.3

Change the $\pm$ to $+$ because sine is positive in the second quadrant.

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

Step 24.1.2.4

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

Tap for more steps...

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.

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

Step 24.1.2.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 24.1.2.4.3

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

Tap for more steps...

Step 24.1.2.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 24.1.2.4.3.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

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

Step 24.1.2.4.4

Write $1$ as a fraction with a common denominator.

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

Step 24.1.2.4.5

Combine the numerators over the common denominator.

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

Step 24.1.2.4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 24.1.2.4.7

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

Tap for more steps...

Step 24.1.2.4.7.1

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

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

Step 24.1.2.4.7.2

Multiply $2$ by $2$ .

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

Step 24.1.2.4.8

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

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

Step 24.1.2.4.9

Simplify the denominator.

Tap for more steps...

Step 24.1.2.4.9.1

Rewrite $4$ as $2^{2}$ .

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

Step 24.1.2.4.9.2

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

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

Step 24.1.3

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

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

Step 24.2

Simplify terms.

Tap for more steps...

Step 24.2.1

Combine the numerators over the common denominator.

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

Step 24.2.2

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

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

Step 24.2.3

Factor $- 1$ out of $- \sqrt{2 - \sqrt{2}}$ .

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

Step 24.2.4

Factor $- 1$ out of $i \sqrt{2 + \sqrt{2}}$ .

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

Step 24.2.5

Factor $- 1$ out of $- (\sqrt{2 - \sqrt{2}}) - (- i \sqrt{2 + \sqrt{2}})$ .

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

Step 24.2.6

Simplify the expression.

Tap for more steps...

Step 24.2.6.1

Rewrite $- (\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}})$ as $- 1 (\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}})$ .

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

Step 24.2.6.2

Move the negative in front of the fraction.

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

Step 25

Substitute $z^{4}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{1} = 0 - \frac{3 (\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}})}{2}$

Step 26

Find the value of $θ$ for $r = 2$ .

$4 θ = \frac{π}{2} + 2 π (2)$

Step 27

Solve the equation for $θ$ .

Tap for more steps...

Step 27.1

Simplify.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 27.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

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.

Tap for more steps...

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 $\frac{9 π}{2} \cdot \frac{1}{4}$ .

Tap for more steps...

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 $u^{4} = 81 i$ .

$u_{2} = 3 (\cos (\frac{9 π}{8}) + i \sin (\frac{9 π}{8}))$

Step 29

Convert the solution to rectangular form.

Tap for more steps...

Step 29.1

Simplify each term.

Tap for more steps...

Step 29.1.1

The exact value of $\cos (\frac{9 π}{8})$ is $- \frac{\sqrt{2 + \sqrt{2}}}{2}$ .

Tap for more steps...

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 $2$ .

$u_{2} = 3 (\cos (\frac{\frac{9 π}{4}}{2}) + i \sin (\frac{9 π}{8}))$

Step 29.1.1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$u_{2} = 3 (\pm \sqrt{\frac{1 + \cos (\frac{9 π}{4})}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.3

Change the $\pm$ to $-$ because cosine is negative in the third quadrant.

$u_{2} = 3 (- \sqrt{\frac{1 + \cos (\frac{9 π}{4})}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4

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

Tap for more steps...

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 $2 π$ .

$u_{2} = 3 (- \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$u_{2} = 3 (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.3

Write $1$ as a fraction with a common denominator.

$u_{2} = 3 (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.4

Combine the numerators over the common denominator.

$u_{2} = 3 (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

$u_{2} = 3 (- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.6

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

Tap for more steps...

Step 29.1.1.4.6.1

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

$u_{2} = 3 (- \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.6.2

Multiply $2$ by $2$ .

$u_{2} = 3 (- \sqrt{\frac{2 + \sqrt{2}}{4}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.7

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

$u_{2} = 3 (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.8

Simplify the denominator.

Tap for more steps...

Step 29.1.1.4.8.1

Rewrite $4$ as $2^{2}$ .

$u_{2} = 3 (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}} + i \sin (\frac{9 π}{8}))$

Step 29.1.1.4.8.2

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

$u_{2} = 3 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i \sin (\frac{9 π}{8}))$

Step 29.1.2

The exact value of $\sin (\frac{9 π}{8})$ is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ .

Tap for more steps...

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 $2$ .

$u_{2} = 3 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i \sin (\frac{\frac{9 π}{4}}{2}))$

Step 29.1.2.2

Apply the sine half-angle identity.

$u_{2} = 3 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (\pm \sqrt{\frac{1 - \cos (\frac{9 π}{4})}{2}}))$

Step 29.1.2.3

Change the $\pm$ to $-$ because sine is negative in the third quadrant.

$u_{2} = 3 (- \frac{\sqrt{2 + \sqrt{2}}}{2} + i (- \sqrt{\frac{1 - \cos (\frac{9 π}{4})}{2}}))$

Step 29.1.2.4

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

Tap for more steps...

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 $2 π$ .

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

Step 29.1.2.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 29.1.2.4.3

Write $1$ as a fraction with a common denominator.

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

Step 29.1.2.4.4

Combine the numerators over the common denominator.

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

Step 29.1.2.4.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 29.1.2.4.6

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

Tap for more steps...

Step 29.1.2.4.6.1

Multiply $\frac{2 - \sqrt{2}}{2}$ by $\frac{1}{2}$ .

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

Step 29.1.2.4.6.2

Multiply $2$ by $2$ .

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

Step 29.1.2.4.7

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

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

Step 29.1.2.4.8

Simplify the denominator.

Tap for more steps...

Step 29.1.2.4.8.1

Rewrite $4$ as $2^{2}$ .

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

Step 29.1.2.4.8.2

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

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

Step 29.1.3

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

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

Step 29.2

Simplify terms.

Tap for more steps...

Step 29.2.1

Combine the numerators over the common denominator.

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

Step 29.2.2

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

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

Step 29.2.3

Factor $- 1$ out of $- \sqrt{2 + \sqrt{2}}$ .

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

Step 29.2.4

Factor $- 1$ out of $- i \sqrt{2 - \sqrt{2}}$ .

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

Step 29.2.5

Factor $- 1$ out of $- (\sqrt{2 + \sqrt{2}}) - (i \sqrt{2 - \sqrt{2}})$ .

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

Step 29.2.6

Simplify the expression.

Tap for more steps...

Step 29.2.6.1

Rewrite $- (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})$ as $- 1 (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})$ .

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

Step 29.2.6.2

Move the negative in front of the fraction.

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

Step 30

Substitute $z^{4}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{2} = 0 - \frac{3 (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})}{2}$

Step 31

Find the value of $θ$ for $r = 3$ .

$4 θ = \frac{π}{2} + 2 π (3)$

Step 32

Solve the equation for $θ$ .

Tap for more steps...

Step 32.1

Simplify.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 32.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

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.

Tap for more steps...

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 $\frac{13 π}{2} \cdot \frac{1}{4}$ .

Tap for more steps...

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 $u^{4} = 81 i$ .

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

Step 34

Convert the solution to rectangular form.

Tap for more steps...

Step 34.1

Simplify each term.

Tap for more steps...

Step 34.1.1

The exact value of $\cos (\frac{13 π}{8})$ is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

Tap for more steps...

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 $2$ .

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

Step 34.1.1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

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

Step 34.1.1.3

Change the $\pm$ to $+$ because cosine is positive in the fourth quadrant.

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

Step 34.1.1.4

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

Tap for more steps...

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 $2 π$ .

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

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.

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

Step 34.1.1.4.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 34.1.1.4.4

Write $1$ as a fraction with a common denominator.

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

Step 34.1.1.4.5

Combine the numerators over the common denominator.

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

Step 34.1.1.4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 34.1.1.4.7

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

Tap for more steps...

Step 34.1.1.4.7.1

Multiply $\frac{2 - \sqrt{2}}{2}$ by $\frac{1}{2}$ .

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

Step 34.1.1.4.7.2

Multiply $2$ by $2$ .

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

Step 34.1.1.4.8

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

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

Step 34.1.1.4.9

Simplify the denominator.

Tap for more steps...

Step 34.1.1.4.9.1

Rewrite $4$ as $2^{2}$ .

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

Step 34.1.1.4.9.2

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

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

Step 34.1.2

The exact value of $\sin (\frac{13 π}{8})$ is $- \frac{\sqrt{2 + \sqrt{2}}}{2}$ .

Tap for more steps...

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 $2$ .

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

Step 34.1.2.2

Apply the sine half-angle identity.

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

Step 34.1.2.3

Change the $\pm$ to $-$ because sine is negative in the fourth quadrant.

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

Step 34.1.2.4

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

Tap for more steps...

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 $2 π$ .

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

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.

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

Step 34.1.2.4.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 34.1.2.4.4

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

Tap for more steps...

Step 34.1.2.4.4.1

Multiply $- 1$ by $- 1$ .

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

Step 34.1.2.4.4.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

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

Step 34.1.2.4.5

Write $1$ as a fraction with a common denominator.

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

Step 34.1.2.4.6

Combine the numerators over the common denominator.

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

Step 34.1.2.4.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 34.1.2.4.8

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

Tap for more steps...

Step 34.1.2.4.8.1

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

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

Step 34.1.2.4.8.2

Multiply $2$ by $2$ .

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

Step 34.1.2.4.9

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

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

Step 34.1.2.4.10

Simplify the denominator.

Tap for more steps...

Step 34.1.2.4.10.1

Rewrite $4$ as $2^{2}$ .

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

Step 34.1.2.4.10.2

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

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

Step 34.1.3

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

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

Step 34.2

Combine fractions.

Tap for more steps...

Step 34.2.1

Combine the numerators over the common denominator.

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

Step 34.2.2

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

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

Step 35

Substitute $z^{4}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{3} = 0 + \frac{3 (\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}})}{2}$

Step 36

These are the complex solutions to $u^{4} = 81 i$ .

$z_{0} = \frac{3 (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})}{2}$

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

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

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