Algebra Examples

$z^{3} = 27 i$

Step 1

Substitute $u$ for $z^{3}$ .

$u^{3} = 27 i$

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 $b = 27$ .

$| z | = \sqrt{27^{2}}$

Step 5

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

$| z | = 27$

Step 6

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

$θ = \arctan (\frac{27}{0})$

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 $| z | = 27$ .

$27 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 9

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

$u^{3} = 27 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 10

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

$r^{3} (c o s (3 θ) + i s i n (3 θ)) = 27 i = 27 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 11

Equate the modulus of the trigonometric form to $r^{3}$ to find the value of $r$ .

$r^{3} = 27$

Step 12

Solve the equation for $r$ .

Tap for more steps...

Step 12.1

Subtract $27$ from both sides of the equation.

$r^{3} - 27 = 0$

Step 12.2

Factor the left side of the equation.

Tap for more steps...

Step 12.2.1

Rewrite $27$ as $3^{3}$ .

$r^{3} - 3^{3} = 0$

Step 12.2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = r$ and $b = 3$ .

$(r - 3) (r^{2} + r \cdot 3 + 3^{2}) = 0$

Step 12.2.3

Simplify.

Tap for more steps...

Step 12.2.3.1

Move $3$ to the left of $r$ .

$(r - 3) (r^{2} + 3 r + 3^{2}) = 0$

Step 12.2.3.2

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

$(r - 3) (r^{2} + 3 r + 9) = 0$

Step 12.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$r - 3 = 0$

$r^{2} + 3 r + 9 = 0$

Step 12.4

Set $r - 3$ equal to $0$ and solve for $r$ .

Tap for more steps...

Step 12.4.1

Set $r - 3$ equal to $0$ .

$r - 3 = 0$

Step 12.4.2

Add $3$ to both sides of the equation.

$r = 3$

Step 12.5

Set $r^{2} + 3 r + 9$ equal to $0$ and solve for $r$ .

Tap for more steps...

Step 12.5.1

Set $r^{2} + 3 r + 9$ equal to $0$ .

$r^{2} + 3 r + 9 = 0$

Step 12.5.2

Solve $r^{2} + 3 r + 9 = 0$ for $r$ .

Tap for more steps...

Step 12.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 12.5.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 9$ into the quadratic formula and solve for $r$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 12.5.2.3

Simplify.

Tap for more steps...

Step 12.5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 12.5.2.3.1.1

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

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 12.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

Tap for more steps...

Step 12.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 12.5.2.3.1.2.2

Multiply $- 4$ by $9$ .

$r = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 12.5.2.3.1.3

Subtract $36$ from $9$ .

$r = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 12.5.2.3.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$r = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 12.5.2.3.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$r = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 12.5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$r = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 12.5.2.3.1.7

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

Tap for more steps...

Step 12.5.2.3.1.7.1

Factor $9$ out of $27$ .

$r = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 12.5.2.3.1.7.2

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

$r = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 12.5.2.3.1.8

Pull terms out from under the radical.

$r = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 12.5.2.3.1.9

Move $3$ to the left of $i$ .

$r = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 12.5.2.3.2

Multiply $2$ by $1$ .

$r = \frac{- 3 \pm 3 i \sqrt{3}}{2}$

Step 12.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 12.5.2.4.1

Simplify the numerator.

Tap for more steps...

Step 12.5.2.4.1.1

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

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 12.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

Tap for more steps...

Step 12.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 12.5.2.4.1.2.2

Multiply $- 4$ by $9$ .

$r = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 12.5.2.4.1.3

Subtract $36$ from $9$ .

$r = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 12.5.2.4.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$r = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 12.5.2.4.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$r = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 12.5.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$r = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 12.5.2.4.1.7

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

Tap for more steps...

Step 12.5.2.4.1.7.1

Factor $9$ out of $27$ .

$r = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 12.5.2.4.1.7.2

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

$r = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 12.5.2.4.1.8

Pull terms out from under the radical.

$r = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 12.5.2.4.1.9

Move $3$ to the left of $i$ .

$r = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 12.5.2.4.2

Multiply $2$ by $1$ .

$r = \frac{- 3 \pm 3 i \sqrt{3}}{2}$

Step 12.5.2.4.3

Change the $\pm$ to $+$ .

$r = \frac{- 3 + 3 i \sqrt{3}}{2}$

Step 12.5.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 12.5.2.4.5

Factor $- 1$ out of $3 i \sqrt{3}$ .

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

Step 12.5.2.4.6

Factor $- 1$ out of $- 1 (3) - (- 3 i \sqrt{3})$ .

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

Step 12.5.2.4.7

Move the negative in front of the fraction.

$r = - \frac{3 - 3 i \sqrt{3}}{2}$

Step 12.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 12.5.2.5.1

Simplify the numerator.

Tap for more steps...

Step 12.5.2.5.1.1

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

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 12.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

Tap for more steps...

Step 12.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 12.5.2.5.1.2.2

Multiply $- 4$ by $9$ .

$r = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 12.5.2.5.1.3

Subtract $36$ from $9$ .

$r = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 12.5.2.5.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$r = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 12.5.2.5.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$r = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 12.5.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$r = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 12.5.2.5.1.7

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

Tap for more steps...

Step 12.5.2.5.1.7.1

Factor $9$ out of $27$ .

$r = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 12.5.2.5.1.7.2

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

$r = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 12.5.2.5.1.8

Pull terms out from under the radical.

$r = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 12.5.2.5.1.9

Move $3$ to the left of $i$ .

$r = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 12.5.2.5.2

Multiply $2$ by $1$ .

$r = \frac{- 3 \pm 3 i \sqrt{3}}{2}$

Step 12.5.2.5.3

Change the $\pm$ to $-$ .

$r = \frac{- 3 - 3 i \sqrt{3}}{2}$

Step 12.5.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$r = \frac{- 1 \cdot 3 - 3 i \sqrt{3}}{2}$

Step 12.5.2.5.5

Factor $- 1$ out of $- 3 i \sqrt{3}$ .

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

Step 12.5.2.5.6

Factor $- 1$ out of $- 1 (3) - (3 i \sqrt{3})$ .

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

Step 12.5.2.5.7

Move the negative in front of the fraction.

$r = - \frac{3 + 3 i \sqrt{3}}{2}$

Step 12.5.2.6

The final answer is the combination of both solutions.

$r = - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

Step 12.6

The final solution is all the values that make $(r - 3) (r^{2} + 3 r + 9) = 0$ true.

$r = 3, - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

Step 13

Find the approximate value of $r$ .

$r = 3$

Step 14

Find the possible values of $θ$ .

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

Step 15

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

$3 θ = \frac{π}{2} + 2 π n$

Step 16

Find the value of $θ$ for $r = 0$ .

$3 θ = \frac{π}{2} + 2 π (0)$

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 $2$ .

$3 θ = \frac{π}{2} + 0 π$

Step 17.1.1.2

Multiply $0$ by $π$ .

$3 θ = \frac{π}{2} + 0$

Step 17.1.2

Add $\frac{π}{2}$ and $0$ .

$3 θ = \frac{π}{2}$

Step 17.2

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

Tap for more steps...

Step 17.2.1

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

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

Step 17.2.2

Simplify the left side.

Tap for more steps...

Step 17.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 17.2.2.1.1

Cancel the common factor.

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

Step 17.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{2}}{3}$

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.

$θ = \frac{π}{2} \cdot \frac{1}{3}$

Step 17.2.3.2

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

Tap for more steps...

Step 17.2.3.2.1

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

$θ = \frac{π}{2 \cdot 3}$

Step 17.2.3.2.2

Multiply $2$ by $3$ .

$θ = \frac{π}{6}$

Step 18

Use the values of $θ$ and $r$ to find a solution to the equation $u^{3} = 27 i$ .

$u_{0} = 3 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))$

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 $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$u_{0} = 3 (\frac{\sqrt{3}}{2} + i \sin (\frac{π}{6}))$

Step 19.1.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 19.1.3

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

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

Step 19.2

Simplify terms.

Tap for more steps...

Step 19.2.1

Apply the distributive property.

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

Step 19.2.2

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

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

Step 19.2.3

Combine $3$ and $\frac{i}{2}$ .

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

Step 20

Substitute $z^{3}$ for $u$ to calculate the value of $z$ after the right shift.

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

Step 21

Find the value of $θ$ for $r = 1$ .

$3 θ = \frac{π}{2} + 2 π (1)$

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 $1$ .

$3 θ = \frac{π}{2} + 2 π$

Step 22.1.2

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

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

Step 22.1.3

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 22.1.4

Combine the numerators over the common denominator.

$3 θ = \frac{π + 2 π \cdot 2}{2}$

Step 22.1.5

Multiply $2$ by $2$ .

$3 θ = \frac{π + 4 π}{2}$

Step 22.1.6

Add $π$ and $4 π$ .

$3 θ = \frac{5 π}{2}$

Step 22.2

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

Tap for more steps...

Step 22.2.1

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

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

Step 22.2.2

Simplify the left side.

Tap for more steps...

Step 22.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 22.2.2.1.1

Cancel the common factor.

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

Step 22.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{5 π}{2}}{3}$

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.

$θ = \frac{5 π}{2} \cdot \frac{1}{3}$

Step 22.2.3.2

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

Tap for more steps...

Step 22.2.3.2.1

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

$θ = \frac{5 π}{2 \cdot 3}$

Step 22.2.3.2.2

Multiply $2$ by $3$ .

$θ = \frac{5 π}{6}$

Step 23

Use the values of $θ$ and $r$ to find a solution to the equation $u^{3} = 27 i$ .

$u_{1} = 3 (\cos (\frac{5 π}{6}) + i \sin (\frac{5 π}{6}))$

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

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 second quadrant.

$u_{1} = 3 (- \cos (\frac{π}{6}) + i \sin (\frac{5 π}{6}))$

Step 24.1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$u_{1} = 3 (- \frac{\sqrt{3}}{2} + i \sin (\frac{5 π}{6}))$

Step 24.1.3

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

$u_{1} = 3 (- \frac{\sqrt{3}}{2} + i \sin (\frac{π}{6}))$

Step 24.1.4

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 24.1.5

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

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

Step 24.2

Apply the distributive property.

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

Step 24.3

Multiply $3 (- \frac{\sqrt{3}}{2})$ .

Tap for more steps...

Step 24.3.1

Multiply $- 1$ by $3$ .

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

Step 24.3.2

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

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

Step 24.4

Combine $3$ and $\frac{i}{2}$ .

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

Step 24.5

Move the negative in front of the fraction.

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

Step 25

Substitute $z^{3}$ for $u$ to calculate the value of $z$ after the right shift.

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

Step 26

Find the value of $θ$ for $r = 2$ .

$3 θ = \frac{π}{2} + 2 π (2)$

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 $2$ .

$3 θ = \frac{π}{2} + 4 π$

Step 27.1.2

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

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

Step 27.1.3

Combine $4 π$ and $\frac{2}{2}$ .

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

Step 27.1.4

Combine the numerators over the common denominator.

$3 θ = \frac{π + 4 π \cdot 2}{2}$

Step 27.1.5

Multiply $2$ by $4$ .

$3 θ = \frac{π + 8 π}{2}$

Step 27.1.6

Add $π$ and $8 π$ .

$3 θ = \frac{9 π}{2}$

Step 27.2

Divide each term in $3 θ = \frac{9 π}{2}$ by $3$ and simplify.

Tap for more steps...

Step 27.2.1

Divide each term in $3 θ = \frac{9 π}{2}$ by $3$ .

$\frac{3 θ}{3} = \frac{\frac{9 π}{2}}{3}$

Step 27.2.2

Simplify the left side.

Tap for more steps...

Step 27.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 27.2.2.1.1

Cancel the common factor.

$\frac{3 θ}{3} = \frac{\frac{9 π}{2}}{3}$

Step 27.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{9 π}{2}}{3}$

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.

$θ = \frac{9 π}{2} \cdot \frac{1}{3}$

Step 27.2.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 27.2.3.2.1

Factor $3$ out of $9 π$ .

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

Step 27.2.3.2.2

Cancel the common factor.

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

Step 27.2.3.2.3

Rewrite the expression.

$θ = \frac{3 π}{2}$

Step 28

Use the values of $θ$ and $r$ to find a solution to the equation $u^{3} = 27 i$ .

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

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

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

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

Step 29.1.2

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

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

Step 29.1.3

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.

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

Step 29.1.4

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

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

Step 29.1.5

Multiply $- 1$ by $1$ .

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

Step 29.1.6

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

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

Step 29.1.7

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

$u_{2} = 3 (0 - i)$

Step 29.2

Simplify the expression.

Tap for more steps...

Step 29.2.1

Subtract $i$ from $0$ .

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

Step 29.2.2

Multiply $- 1$ by $3$ .

$u_{2} = - 3 i$

Step 30

Substitute $z^{3}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{2} = 0 - 3 i$

Step 31

These are the complex solutions to $u^{3} = 27 i$ .

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

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

$z_{2} = - 3 i$