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z^4=1

z^4=1 equation

A equation with variable:

Numerical solution:

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The solution

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 4    
z  = 1
$$z^{4} = 1$$
Detail solution
Given the equation
$$z^{4} = 1$$
Because equation degree is equal to = 4 - contains the even number 4 in the numerator, then
the equation has two real roots.
Get the root 4-th degree of the equation sides:
We get:
$$\sqrt[4]{z^{4}} = \sqrt[4]{1}$$
$$\sqrt[4]{z^{4}} = \left(-1\right) \sqrt[4]{1}$$
or
$$z = 1$$
$$z = -1$$
We get the answer: z = 1
We get the answer: z = -1
or
$$z_{1} = -1$$
$$z_{2} = 1$$

All other 2 root(s) is the complex numbers.
do replacement:
$$w = z$$
then the equation will be the:
$$w^{4} = 1$$
Any complex number can presented so:
$$w = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = 1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{4 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = 1$$
so
$$\cos{\left(4 p \right)} = 1$$
and
$$\sin{\left(4 p \right)} = 0$$
then
$$p = \frac{\pi N}{2}$$
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:
$$w_{1} = -1$$
$$w_{2} = 1$$
$$w_{3} = - i$$
$$w_{4} = i$$
do backward replacement
$$w = z$$
$$z = w$$

The final answer:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{3} = - i$$
$$z_{4} = i$$
The graph
Rapid solution [src]
z1 = -1
$$z_{1} = -1$$
z2 = 1
$$z_{2} = 1$$
z3 = -I
$$z_{3} = - i$$
z4 = I
$$z_{4} = i$$
z4 = i
Sum and product of roots [src]
sum
-1 + 1 - I + I
$$\left(\left(-1 + 1\right) - i\right) + i$$
=
0
$$0$$
product
-(-I)*I
$$i \left(- \left(-1\right) i\right)$$
=
-1
$$-1$$
-1
Numerical answer [src]
z1 = 1.0
z2 = 1.0*i
z3 = -1.0
z4 = -1.0*i
z4 = -1.0*i
The graph
z^4=1 equation
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