Algebra Examples

$x^{4} = - 2 + 2 \sqrt{3} i$

Step 1

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

$u^{4} = - 2 + 2 \sqrt{3} i$

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 $a = - 2$ and $b = 2 \sqrt{3}$ .

$| z | = \sqrt{{(2 \sqrt{3})}^{2} + {(- 2)}^{2}}$

Step 5

Find $| z |$ .

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Step 5.1

Simplify the expression.

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Step 5.1.1

Apply the product rule to $2 \sqrt{3}$ .

$| z | = \sqrt{2^{2} {\sqrt{3}}^{2} + {(- 2)}^{2}}$

Step 5.1.2

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

$| z | = \sqrt{4 {\sqrt{3}}^{2} + {(- 2)}^{2}}$

Step 5.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$| z | = \sqrt{4 {(3^{\frac{1}{2}})}^{2} + {(- 2)}^{2}}$

Step 5.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \sqrt{4 \cdot 3^{\frac{1}{2} \cdot 2} + {(- 2)}^{2}}$

Step 5.2.3

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

$| z | = \sqrt{4 \cdot 3^{\frac{2}{2}} + {(- 2)}^{2}}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$| z | = \sqrt{4 \cdot 3^{\frac{2}{2}} + {(- 2)}^{2}}$

Step 5.2.4.2

Rewrite the expression.

$| z | = \sqrt{4 \cdot 3 + {(- 2)}^{2}}$

Step 5.2.5

Evaluate the exponent.

$| z | = \sqrt{4 \cdot 3 + {(- 2)}^{2}}$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $4$ by $3$ .

$| z | = \sqrt{12 + {(- 2)}^{2}}$

Step 5.3.2

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

$| z | = \sqrt{12 + 4}$

Step 5.3.3

Add $12$ and $4$ .

$| z | = \sqrt{16}$

Step 5.3.4

Rewrite $16$ as $4^{2}$ .

$| z | = \sqrt{4^{2}}$

Step 5.3.5

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

$| z | = 4$

Step 6

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

$θ = \arctan (\frac{2 \sqrt{3}}{- 2})$

Step 7

Since inverse tangent of $\frac{2 \sqrt{3}}{- 2}$ produces an angle in the second quadrant, the value of the angle is $\frac{2 π}{3}$ .

$θ = \frac{2 π}{3}$

Step 8

Substitute the values of $θ = \frac{2 π}{3}$ and $| z | = 4$ .

$4 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3}))$

Step 9

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

$u^{4} = 4 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3}))$

Step 10

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

$r^{4} (c o s (4 θ) + i s i n (4 θ)) = - 2 + 2 \sqrt{3} i = 4 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3}))$

Step 11

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

$r^{4} = 4$

Step 12

Solve the equation for $r$ .

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Step 12.1

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

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

Step 12.2

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

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Step 12.2.1

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

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

Step 12.2.2

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

$r = \pm \sqrt{\sqrt{2^{2}}}$

Step 12.2.3

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

$r = \pm \sqrt{2}$

Step 12.3

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

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Step 12.3.1

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

$r = \sqrt{2}$

Step 12.3.2

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

$r = - \sqrt{2}$

Step 12.3.3

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

$r = \sqrt{2}, - \sqrt{2}$

Step 13

Find the approximate value of $r$ .

$r = 1.41421356$

Step 14

Find the possible values of $θ$ .

$c o s (4 θ) = c o s (\frac{2 π}{3} + 2 π n)$ and $s i n (4 θ) = s i n (\frac{2 π}{3} + 2 π n)$

Step 15

Finding all the possible values of $θ$ leads to the equation $4 θ = \frac{2 π}{3} + 2 π n$ .

$4 θ = \frac{2 π}{3} + 2 π n$

Step 16

Find the value of $θ$ for $r = 0$ .

$4 θ = \frac{2 π}{3} + 2 π (0)$

Step 17

Solve the equation for $θ$ .

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Step 17.1

Simplify.

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Step 17.1.1

Multiply $2 π \cdot 0$ .

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Step 17.1.1.1

Multiply $0$ by $2$ .

$4 θ = \frac{2 π}{3} + 0 π$

Step 17.1.1.2

Multiply $0$ by $π$ .

$4 θ = \frac{2 π}{3} + 0$

Step 17.1.2

Add $\frac{2 π}{3}$ and $0$ .

$4 θ = \frac{2 π}{3}$

Step 17.2

Divide each term in $4 θ = \frac{2 π}{3}$ by $4$ and simplify.

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Step 17.2.1

Divide each term in $4 θ = \frac{2 π}{3}$ by $4$ .

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

Step 17.2.2

Simplify the left side.

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Step 17.2.2.1

Cancel the common factor of $4$ .

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Step 17.2.2.1.1

Cancel the common factor.

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

Step 17.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{2 π}{3}}{4}$

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.

$θ = \frac{2 π}{3} \cdot \frac{1}{4}$

Step 17.2.3.2

Cancel the common factor of $2$ .

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Step 17.2.3.2.1

Factor $2$ out of $2 π$ .

$θ = \frac{2 (π)}{3} \cdot \frac{1}{4}$

Step 17.2.3.2.2

Factor $2$ out of $4$ .

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

Step 17.2.3.2.3

Cancel the common factor.

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

Step 17.2.3.2.4

Rewrite the expression.

$θ = \frac{π}{3} \cdot \frac{1}{2}$

Step 17.2.3.3

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

$θ = \frac{π}{3 \cdot 2}$

Step 17.2.3.4

Multiply $3$ by $2$ .

$θ = \frac{π}{6}$

Step 18

Use the values of $θ$ and $r$ to find a solution to the equation $u^{4} = - 2 + 2 \sqrt{3} i$ .

$u_{0} = 1.41421356 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))$

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

$u_{0} = 1.41421356 (\frac{\sqrt{3}}{2} + i \sin (\frac{π}{6}))$

Step 19.1.2

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

$u_{0} = 1.41421356 (\frac{\sqrt{3}}{2} + i (\frac{1}{2}))$

Step 19.1.3

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

$u_{0} = 1.41421356 (\frac{\sqrt{3}}{2} + \frac{i}{2})$

Step 19.2

Apply the distributive property.

$u_{0} = 1.41421356 (\frac{\sqrt{3}}{2}) + 1.41421356 (\frac{i}{2})$

Step 19.3

Multiply $1.41421356 \frac{\sqrt{3}}{2}$ .

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Step 19.3.1

Combine $1.41421356$ and $\frac{\sqrt{3}}{2}$ .

$u_{0} = \frac{1.41421356 \sqrt{3}}{2} + 1.41421356 (\frac{i}{2})$

Step 19.3.2

Multiply $1.41421356$ by $\sqrt{3}$ .

$u_{0} = \frac{2.44948974}{2} + 1.41421356 (\frac{i}{2})$

Step 19.4

Combine $1.41421356$ and $\frac{i}{2}$ .

$u_{0} = \frac{2.44948974}{2} + \frac{1.41421356 i}{2}$

Step 19.5

Simplify each term.

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Step 19.5.1

Divide $2.44948974$ by $2$ .

$u_{0} = 1.22474487 + \frac{1.41421356 i}{2}$

Step 19.5.2

Factor $1.41421356$ out of $1.41421356 i$ .

$u_{0} = 1.22474487 + \frac{1.41421356 (i)}{2}$

Step 19.5.3

Factor $2$ out of $2$ .

$u_{0} = 1.22474487 + \frac{1.41421356 (i)}{2 (1)}$

Step 19.5.4

Separate fractions.

$u_{0} = 1.22474487 + \frac{1.41421356}{2} \cdot \frac{i}{1}$

Step 19.5.5

Divide $1.41421356$ by $2$ .

$u_{0} = 1.22474487 + 0.70710678 (\frac{i}{1})$

Step 19.5.6

Divide $i$ by $1$ .

$u_{0} = 1.22474487 + 0.70710678 i$

Step 20

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

$z_{0} = 0 + 1.22474487 + 0.70710678 i$

Step 21

Find the value of $θ$ for $r = 1$ .

$4 θ = \frac{2 π}{3} + 2 π (1)$

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 $1$ .

$4 θ = \frac{2 π}{3} + 2 π$

Step 22.1.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 22.1.3

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 22.1.4

Combine the numerators over the common denominator.

$4 θ = \frac{2 π + 2 π \cdot 3}{3}$

Step 22.1.5

Multiply $3$ by $2$ .

$4 θ = \frac{2 π + 6 π}{3}$

Step 22.1.6

Add $2 π$ and $6 π$ .

$4 θ = \frac{8 π}{3}$

Step 22.2

Divide each term in $4 θ = \frac{8 π}{3}$ by $4$ and simplify.

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Step 22.2.1

Divide each term in $4 θ = \frac{8 π}{3}$ by $4$ .

$\frac{4 θ}{4} = \frac{\frac{8 π}{3}}{4}$

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.

$\frac{4 θ}{4} = \frac{\frac{8 π}{3}}{4}$

Step 22.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{8 π}{3}}{4}$

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.

$θ = \frac{8 π}{3} \cdot \frac{1}{4}$

Step 22.2.3.2

Cancel the common factor of $4$ .

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Step 22.2.3.2.1

Factor $4$ out of $8 π$ .

$θ = \frac{4 (2 π)}{3} \cdot \frac{1}{4}$

Step 22.2.3.2.2

Cancel the common factor.

$θ = \frac{4 (2 π)}{3} \cdot \frac{1}{4}$

Step 22.2.3.2.3

Rewrite the expression.

$θ = \frac{2 π}{3}$

Step 23

Use the values of $θ$ and $r$ to find a solution to the equation $u^{4} = - 2 + 2 \sqrt{3} i$ .

$u_{1} = 1.41421356 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3}))$

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

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.

$u_{1} = 1.41421356 (- \cos (\frac{π}{3}) + i \sin (\frac{2 π}{3}))$

Step 24.1.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{1} = 1.41421356 (- \frac{1}{2} + i \sin (\frac{2 π}{3}))$

Step 24.1.3

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

$u_{1} = 1.41421356 (- \frac{1}{2} + i \sin (\frac{π}{3}))$

Step 24.1.4

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

$u_{1} = 1.41421356 (- \frac{1}{2} + i (\frac{\sqrt{3}}{2}))$

Step 24.1.5

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

$u_{1} = 1.41421356 (- \frac{1}{2} + \frac{i \sqrt{3}}{2})$

Step 24.2

Apply the distributive property.

$u_{1} = 1.41421356 (- \frac{1}{2}) + 1.41421356 (\frac{i \sqrt{3}}{2})$

Step 24.3

Multiply $1.41421356 (- \frac{1}{2})$ .

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Step 24.3.1

Multiply $- 1$ by $1.41421356$ .

$u_{1} = - 1.41421356 (\frac{1}{2}) + 1.41421356 (\frac{i \sqrt{3}}{2})$

Step 24.3.2

Combine $- 1.41421356$ and $\frac{1}{2}$ .

$u_{1} = \frac{- 1.41421356}{2} + 1.41421356 (\frac{i \sqrt{3}}{2})$

Step 24.4

Multiply $1.41421356 \frac{i \sqrt{3}}{2}$ .

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Step 24.4.1

Combine $1.41421356$ and $\frac{i \sqrt{3}}{2}$ .

$u_{1} = \frac{- 1.41421356}{2} + \frac{1.41421356 (i \sqrt{3})}{2}$

Step 24.4.2

Multiply $\sqrt{3}$ by $1.41421356$ .

$u_{1} = \frac{- 1.41421356}{2} + \frac{2.44948974 i}{2}$

Step 24.5

Simplify each term.

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Step 24.5.1

Divide $- 1.41421356$ by $2$ .

$u_{1} = - 0.70710678 + \frac{2.44948974 i}{2}$

Step 24.5.2

Factor $2.44948974$ out of $2.44948974 i$ .

$u_{1} = - 0.70710678 + \frac{2.44948974 (i)}{2}$

Step 24.5.3

Factor $2$ out of $2$ .

$u_{1} = - 0.70710678 + \frac{2.44948974 (i)}{2 (1)}$

Step 24.5.4

Separate fractions.

$u_{1} = - 0.70710678 + \frac{2.44948974}{2} \cdot \frac{i}{1}$

Step 24.5.5

Divide $2.44948974$ by $2$ .

$u_{1} = - 0.70710678 + 1.22474487 (\frac{i}{1})$

Step 24.5.6

Divide $i$ by $1$ .

$u_{1} = - 0.70710678 + 1.22474487 i$

Step 25

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

$z_{1} = 0 - 0.70710678 + 1.22474487 i$

Step 26

Find the value of $θ$ for $r = 2$ .

$4 θ = \frac{2 π}{3} + 2 π (2)$

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 $2$ .

$4 θ = \frac{2 π}{3} + 4 π$

Step 27.1.2

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 27.1.3

Combine $4 π$ and $\frac{3}{3}$ .

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

Step 27.1.4

Combine the numerators over the common denominator.

$4 θ = \frac{2 π + 4 π \cdot 3}{3}$

Step 27.1.5

Multiply $3$ by $4$ .

$4 θ = \frac{2 π + 12 π}{3}$

Step 27.1.6

Add $2 π$ and $12 π$ .

$4 θ = \frac{14 π}{3}$

Step 27.2

Divide each term in $4 θ = \frac{14 π}{3}$ by $4$ and simplify.

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Step 27.2.1

Divide each term in $4 θ = \frac{14 π}{3}$ by $4$ .

$\frac{4 θ}{4} = \frac{\frac{14 π}{3}}{4}$

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.

$\frac{4 θ}{4} = \frac{\frac{14 π}{3}}{4}$

Step 27.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{14 π}{3}}{4}$

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.

$θ = \frac{14 π}{3} \cdot \frac{1}{4}$

Step 27.2.3.2

Cancel the common factor of $2$ .

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Step 27.2.3.2.1

Factor $2$ out of $14 π$ .

$θ = \frac{2 (7 π)}{3} \cdot \frac{1}{4}$

Step 27.2.3.2.2

Factor $2$ out of $4$ .

$θ = \frac{2 (7 π)}{3} \cdot \frac{1}{2 (2)}$

Step 27.2.3.2.3

Cancel the common factor.

$θ = \frac{2 (7 π)}{3} \cdot \frac{1}{2 \cdot 2}$

Step 27.2.3.2.4

Rewrite the expression.

$θ = \frac{7 π}{3} \cdot \frac{1}{2}$

Step 27.2.3.3

Multiply $\frac{7 π}{3}$ by $\frac{1}{2}$ .

$θ = \frac{7 π}{3 \cdot 2}$

Step 27.2.3.4

Multiply $3$ by $2$ .

$θ = \frac{7 π}{6}$

Step 28

Use the values of $θ$ and $r$ to find a solution to the equation $u^{4} = - 2 + 2 \sqrt{3} i$ .

$u_{2} = 1.41421356 (\cos (\frac{7 π}{6}) + i \sin (\frac{7 π}{6}))$

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

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.

$u_{2} = 1.41421356 (- \cos (\frac{π}{6}) + i \sin (\frac{7 π}{6}))$

Step 29.1.2

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

$u_{2} = 1.41421356 (- \frac{\sqrt{3}}{2} + i \sin (\frac{7 π}{6}))$

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 third quadrant.

$u_{2} = 1.41421356 (- \frac{\sqrt{3}}{2} + i (- \sin (\frac{π}{6})))$

Step 29.1.4

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

$u_{2} = 1.41421356 (- \frac{\sqrt{3}}{2} + i (- \frac{1}{2}))$

Step 29.1.5

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

$u_{2} = 1.41421356 (- \frac{\sqrt{3}}{2} - \frac{i}{2})$

Step 29.2

Apply the distributive property.

$u_{2} = 1.41421356 (- \frac{\sqrt{3}}{2}) + 1.41421356 (- \frac{i}{2})$

Step 29.3

Multiply $1.41421356 (- \frac{\sqrt{3}}{2})$ .

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Step 29.3.1

Multiply $- 1$ by $1.41421356$ .

$u_{2} = - 1.41421356 \frac{\sqrt{3}}{2} + 1.41421356 (- \frac{i}{2})$

Step 29.3.2

Combine $- 1.41421356$ and $\frac{\sqrt{3}}{2}$ .

$u_{2} = \frac{- 1.41421356 \sqrt{3}}{2} + 1.41421356 (- \frac{i}{2})$

Step 29.3.3

Multiply $- 1.41421356$ by $\sqrt{3}$ .

$u_{2} = \frac{- 2.44948974}{2} + 1.41421356 (- \frac{i}{2})$

Step 29.4

Multiply $1.41421356 (- \frac{i}{2})$ .

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Step 29.4.1

Multiply $- 1$ by $1.41421356$ .

$u_{2} = \frac{- 2.44948974}{2} - 1.41421356 \frac{i}{2}$

Step 29.4.2

Combine $- 1.41421356$ and $\frac{i}{2}$ .

$u_{2} = \frac{- 2.44948974}{2} + \frac{- 1.41421356 i}{2}$

Step 29.5

Simplify each term.

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Step 29.5.1

Divide $- 2.44948974$ by $2$ .

$u_{2} = - 1.22474487 + \frac{- 1.41421356 i}{2}$

Step 29.5.2

Move the negative in front of the fraction.

$u_{2} = - 1.22474487 - \frac{(1.41421356) i}{2}$

Step 29.5.3

Factor $1.41421356$ out of $1.41421356 i$ .

$u_{2} = - 1.22474487 - \frac{1.41421356 (i)}{2}$

Step 29.5.4

Factor $2$ out of $2$ .

$u_{2} = - 1.22474487 - \frac{1.41421356 (i)}{2 (1)}$

Step 29.5.5

Separate fractions.

$u_{2} = - 1.22474487 - (\frac{1.41421356}{2} \cdot \frac{i}{1})$

Step 29.5.6

Divide $1.41421356$ by $2$ .

$u_{2} = - 1.22474487 - (0.70710678 (\frac{i}{1}))$

Step 29.5.7

Divide $i$ by $1$ .

$u_{2} = - 1.22474487 - (0.70710678 i)$

Step 29.5.8

Multiply $0.70710678$ by $- 1$ .

$u_{2} = - 1.22474487 - 0.70710678 i$

Step 30

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

$z_{2} = 0 - 1.22474487 - 0.70710678 i$

Step 31

Find the value of $θ$ for $r = 3$ .

$4 θ = \frac{2 π}{3} + 2 π (3)$

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 $2$ .

$4 θ = \frac{2 π}{3} + 6 π$

Step 32.1.2

To write $6 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 θ = \frac{2 π}{3} + 6 π \cdot \frac{3}{3}$

Step 32.1.3

Combine $6 π$ and $\frac{3}{3}$ .

$4 θ = \frac{2 π}{3} + \frac{6 π \cdot 3}{3}$

Step 32.1.4

Combine the numerators over the common denominator.

$4 θ = \frac{2 π + 6 π \cdot 3}{3}$

Step 32.1.5

Multiply $3$ by $6$ .

$4 θ = \frac{2 π + 18 π}{3}$

Step 32.1.6

Add $2 π$ and $18 π$ .

$4 θ = \frac{20 π}{3}$

Step 32.2

Divide each term in $4 θ = \frac{20 π}{3}$ by $4$ and simplify.

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Step 32.2.1

Divide each term in $4 θ = \frac{20 π}{3}$ by $4$ .

$\frac{4 θ}{4} = \frac{\frac{20 π}{3}}{4}$

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.

$\frac{4 θ}{4} = \frac{\frac{20 π}{3}}{4}$

Step 32.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{20 π}{3}}{4}$

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.

$θ = \frac{20 π}{3} \cdot \frac{1}{4}$

Step 32.2.3.2

Cancel the common factor of $4$ .

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Step 32.2.3.2.1

Factor $4$ out of $20 π$ .

$θ = \frac{4 (5 π)}{3} \cdot \frac{1}{4}$

Step 32.2.3.2.2

Cancel the common factor.

$θ = \frac{4 (5 π)}{3} \cdot \frac{1}{4}$

Step 32.2.3.2.3

Rewrite the expression.

$θ = \frac{5 π}{3}$

Step 33

Use the values of $θ$ and $r$ to find a solution to the equation $u^{4} = - 2 + 2 \sqrt{3} i$ .

$u_{3} = 1.41421356 (\cos (\frac{5 π}{3}) + i \sin (\frac{5 π}{3}))$

Step 34

Convert the solution to rectangular form.

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Step 34.1

Simplify each term.

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Step 34.1.1

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

$u_{3} = 1.41421356 (\cos (\frac{π}{3}) + i \sin (\frac{5 π}{3}))$

Step 34.1.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{3} = 1.41421356 (\frac{1}{2} + i \sin (\frac{5 π}{3}))$

Step 34.1.3

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

$u_{3} = 1.41421356 (\frac{1}{2} + i (- \sin (\frac{π}{3})))$

Step 34.1.4

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

$u_{3} = 1.41421356 (\frac{1}{2} + i (- \frac{\sqrt{3}}{2}))$

Step 34.1.5

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

$u_{3} = 1.41421356 (\frac{1}{2} - \frac{i \sqrt{3}}{2})$

Step 34.2

Simplify terms.

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Step 34.2.1

Apply the distributive property.

$u_{3} = 1.41421356 (\frac{1}{2}) + 1.41421356 (- \frac{i \sqrt{3}}{2})$

Step 34.2.2

Combine $1.41421356$ and $\frac{1}{2}$ .

$u_{3} = \frac{1.41421356}{2} + 1.41421356 (- \frac{i \sqrt{3}}{2})$

Step 34.3

Multiply $1.41421356 (- \frac{i \sqrt{3}}{2})$ .

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Step 34.3.1

Multiply $- 1$ by $1.41421356$ .

$u_{3} = \frac{1.41421356}{2} - 1.41421356 \frac{i \sqrt{3}}{2}$

Step 34.3.2

Combine $- 1.41421356$ and $\frac{i \sqrt{3}}{2}$ .

$u_{3} = \frac{1.41421356}{2} + \frac{- 1.41421356 (i \sqrt{3})}{2}$

Step 34.3.3

Multiply $\sqrt{3}$ by $- 1.41421356$ .

$u_{3} = \frac{1.41421356}{2} + \frac{- 2.44948974 i}{2}$

Step 34.4

Simplify each term.

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Step 34.4.1

Divide $1.41421356$ by $2$ .

$u_{3} = 0.70710678 + \frac{- 2.44948974 i}{2}$

Step 34.4.2

Move the negative in front of the fraction.

$u_{3} = 0.70710678 - \frac{(2.44948974) i}{2}$

Step 34.4.3

Factor $2.44948974$ out of $2.44948974 i$ .

$u_{3} = 0.70710678 - \frac{2.44948974 (i)}{2}$

Step 34.4.4

Factor $2$ out of $2$ .

$u_{3} = 0.70710678 - \frac{2.44948974 (i)}{2 (1)}$

Step 34.4.5

Separate fractions.

$u_{3} = 0.70710678 - (\frac{2.44948974}{2} \cdot \frac{i}{1})$

Step 34.4.6

Divide $2.44948974$ by $2$ .

$u_{3} = 0.70710678 - (1.22474487 (\frac{i}{1}))$

Step 34.4.7

Divide $i$ by $1$ .

$u_{3} = 0.70710678 - (1.22474487 i)$

Step 34.4.8

Multiply $1.22474487$ by $- 1$ .

$u_{3} = 0.70710678 - 1.22474487 i$

Step 35

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

$z_{3} = 0 + 0.70710678 - 1.22474487 i$

Step 36

These are the complex solutions to $u^{4} = - 2 + 2 \sqrt{3} i$ .

$z_{0} = 1.22474487 + 0.70710678 i$

$z_{1} = - 0.70710678 + 1.22474487 i$

$z_{2} = - 1.22474487 - 0.70710678 i$

$z_{3} = 0.70710678 - 1.22474487 i$