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x^4+81=0 equation

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Numerical solution:

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The solution

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 4         
x  + 81 = 0
$$x^{4} + 81 = 0$$
Detail solution
Given the equation
$$x^{4} + 81 = 0$$
Because equation degree is equal to = 4 and the free term = -81 < 0,
so the real solutions of the equation d'not exist

All other 4 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{4} = -81$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = -81$$
where
$$r = 3$$
- 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
$$p = \frac{\pi N}{2} + \frac{\pi}{4}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$z_{2} = - \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
$$z_{3} = \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$z_{4} = \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$

The final answer:
$$x_{1} = - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$x_{2} = - \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
$$x_{3} = \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$x_{4} = \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
Rapid solution [src]
           ___         ___
       3*\/ 2    3*I*\/ 2 
x1 = - ------- - ---------
          2          2    
$$x_{1} = - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
           ___         ___
       3*\/ 2    3*I*\/ 2 
x2 = - ------- + ---------
          2          2    
$$x_{2} = - \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
         ___         ___
     3*\/ 2    3*I*\/ 2 
x3 = ------- - ---------
        2          2    
$$x_{3} = \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
         ___         ___
     3*\/ 2    3*I*\/ 2 
x4 = ------- + ---------
        2          2    
$$x_{4} = \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
Sum and product of roots [src]
sum
          ___         ___         ___         ___       ___         ___       ___         ___
      3*\/ 2    3*I*\/ 2      3*\/ 2    3*I*\/ 2    3*\/ 2    3*I*\/ 2    3*\/ 2    3*I*\/ 2 
0 + - ------- - --------- + - ------- + --------- + ------- - --------- + ------- + ---------
         2          2            2          2          2          2          2          2    
$$\left(\left(\frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}\right) - 3 \sqrt{2}\right) + \left(\frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}\right)$$
=
0
$$0$$
product
  /      ___         ___\ /      ___         ___\ /    ___         ___\ /    ___         ___\
  |  3*\/ 2    3*I*\/ 2 | |  3*\/ 2    3*I*\/ 2 | |3*\/ 2    3*I*\/ 2 | |3*\/ 2    3*I*\/ 2 |
1*|- ------- - ---------|*|- ------- + ---------|*|------- - ---------|*|------- + ---------|
  \     2          2    / \     2          2    / \   2          2    / \   2          2    /
$$1 \left(- \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}\right) \left(- \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}\right) \left(\frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}\right) \left(\frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}\right)$$
=
81
$$81$$
81
Numerical answer [src]
x1 = -2.12132034355964 + 2.12132034355964*i
x2 = 2.12132034355964 + 2.12132034355964*i
x3 = -2.12132034355964 - 2.12132034355964*i
x4 = 2.12132034355964 - 2.12132034355964*i
x4 = 2.12132034355964 - 2.12132034355964*i