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Equation:
Equation z^4=i
Equation x⁴-5x²+4=0
Equation √y^22/16y^16
Equation 7+5x-2x^2=0
Express {x} in terms of y where:
(x-17)^2+(y+26)^2=16(x−17)2+(y+26)2=16.
5*x-5*y=20
8*x-7*y=-13
9*x+18*y=6
Derivative of:
z^4
Graphing y =:
z^4
z^4
Identical expressions
z^ four =i
z to the power of 4 equally i
z to the power of four equally i
z4=i
z⁴=i
Similar expressions
z^(4)-z^(3)+z-1=0

z^4=i equation

The teacher will be very surprised to see your correct solution 😉

The solution

You have entered [src]

 4    
z  = I

z^{4} = i

Simplify

Detail solution

Given the equation

z^{4} = i

Because equation degree is equal to = 4 and the free term = i complex,
so the real solutions of the equation d'not exist

All other 4 root(s) is the complex numbers.
do replacement:

w = z

then the equation will be the:

w^{4} = i

Any complex number can presented so:

w = r e^{i p}

substitute to the equation

r^{4} e^{4 i p} = i

where

r = 1

- the magnitude of the complex number
Substitute r:

e^{4 i p} = i

Using Euler’s formula, we find roots for p

i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = i

\cos{\left(4 p \right)} = 0

and

\sin{\left(4 p \right)} = 1

then

p = \frac{\pi N}{2} + \frac{\pi}{8}

where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for w
Consequently, the solution will be for w:

w_{1} = - \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

w_{2} = \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

w_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}

w_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}

do backward replacement

w = z

z = w

The final answer:

z_{1} = - \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

z_{2} = \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

z_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}

z_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}

The graph

Sum and product of roots [src]

sum

     ___________        ___________      ___________        ___________        ___________        ___________      ___________        ___________
    /       ___        /       ___      /       ___        /       ___        /       ___        /       ___      /       ___        /       ___ 
  \/  2 - \/ 2     I*\/  2 + \/ 2     \/  2 - \/ 2     I*\/  2 + \/ 2       \/  2 + \/ 2     I*\/  2 - \/ 2     \/  2 + \/ 2     I*\/  2 - \/ 2  
- -------------- + ---------------- + -------------- - ---------------- + - -------------- - ---------------- + -------------- + ----------------
        2                 2                 2                 2                   2                 2                 2                 2

\left(- \frac{\sqrt{- \sqrt{2} + 2}}{2} + \frac{i \sqrt{\sqrt{2} + 2}}{2}\right) + \left(\frac{\sqrt{- \sqrt{2} + 2}}{2} - \frac{i \sqrt{\sqrt{2} + 2}}{2}\right) + \left(- \frac{\sqrt{\sqrt{2} + 2}}{2} - \frac{i \sqrt{- \sqrt{2} + 2}}{2}\right) + \left(\frac{\sqrt{\sqrt{2} + 2}}{2} + \frac{i \sqrt{- \sqrt{2} + 2}}{2}\right)

0

Simplify

product

     ___________        ___________      ___________        ___________        ___________        ___________      ___________        ___________
    /       ___        /       ___      /       ___        /       ___        /       ___        /       ___      /       ___        /       ___ 
  \/  2 - \/ 2     I*\/  2 + \/ 2     \/  2 - \/ 2     I*\/  2 + \/ 2       \/  2 + \/ 2     I*\/  2 - \/ 2     \/  2 + \/ 2     I*\/  2 - \/ 2  
- -------------- + ---------------- * -------------- - ---------------- * - -------------- - ---------------- * -------------- + ----------------
        2                 2                 2                 2                   2                 2                 2                 2

\left(- \frac{\sqrt{- \sqrt{2} + 2}}{2} + \frac{i \sqrt{\sqrt{2} + 2}}{2}\right) * \left(\frac{\sqrt{- \sqrt{2} + 2}}{2} - \frac{i \sqrt{\sqrt{2} + 2}}{2}\right) * \left(- \frac{\sqrt{\sqrt{2} + 2}}{2} - \frac{i \sqrt{- \sqrt{2} + 2}}{2}\right) * \left(\frac{\sqrt{\sqrt{2} + 2}}{2} + \frac{i \sqrt{- \sqrt{2} + 2}}{2}\right)

-I

- i

Simplify

Rapid solution [src]

           ___________        ___________
          /       ___        /       ___ 
        \/  2 - \/ 2     I*\/  2 + \/ 2  
z_1 = - -------------- + ----------------
              2                 2

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

Simplify

         ___________        ___________
        /       ___        /       ___ 
      \/  2 - \/ 2     I*\/  2 + \/ 2  
z_2 = -------------- - ----------------
            2                 2

z_{2} = \frac{\sqrt{- \sqrt{2} + 2}}{2} - \frac{i \sqrt{\sqrt{2} + 2}}{2}

Simplify

           ___________        ___________
          /       ___        /       ___ 
        \/  2 + \/ 2     I*\/  2 - \/ 2  
z_3 = - -------------- - ----------------
              2                 2

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

Simplify

         ___________        ___________
        /       ___        /       ___ 
      \/  2 + \/ 2     I*\/  2 - \/ 2  
z_4 = -------------- + ----------------
            2                 2

z_{4} = \frac{\sqrt{\sqrt{2} + 2}}{2} + \frac{i \sqrt{- \sqrt{2} + 2}}{2}

Simplify

Numerical answer [src]

z1 = -0.38268343236509 + 0.923879532511287*i

z2 = -0.923879532511287 - 0.38268343236509*i

z3 = 0.38268343236509 - 0.923879532511287*i

z4 = 0.923879532511287 + 0.38268343236509*i

z4 = 0.923879532511287 + 0.38268343236509*i

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z^4=i equation

Numerical solution:

The solution