Mister Exam

# z^4=i equation

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#### Numerical solution:

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### The solution

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 4
z  = I
$$z^{4} = i$$
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
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = i$$
so
$$\cos{\left(4 p \right)} = 0$$
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
$$w = z$$
$$z = w$$

$$z_{1} = - \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{2} = \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}$$
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
$$0$$
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
$$- i$$
Rapid solution [src]
           ___________        ___________
/       ___        /       ___
\/  2 - \/ 2     I*\/  2 + \/ 2
z_1 = - -------------- + ----------------
2                 2        
$$z_{1} = - \frac{\sqrt{- \sqrt{2} + 2}}{2} + \frac{i \sqrt{\sqrt{2} + 2}}{2}$$
         ___________        ___________
/       ___        /       ___
\/  2 - \/ 2     I*\/  2 + \/ 2
z_2 = -------------- - ----------------
2                 2        
$$z_{2} = \frac{\sqrt{- \sqrt{2} + 2}}{2} - \frac{i \sqrt{\sqrt{2} + 2}}{2}$$
           ___________        ___________
/       ___        /       ___
\/  2 + \/ 2     I*\/  2 - \/ 2
z_3 = - -------------- - ----------------
2                 2        
$$z_{3} = - \frac{\sqrt{\sqrt{2} + 2}}{2} - \frac{i \sqrt{- \sqrt{2} + 2}}{2}$$
         ___________        ___________
/       ___        /       ___
\/  2 + \/ 2     I*\/  2 - \/ 2
z_4 = -------------- + ----------------
2                 2        
$$z_{4} = \frac{\sqrt{\sqrt{2} + 2}}{2} + \frac{i \sqrt{- \sqrt{2} + 2}}{2}$$
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