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z^5+32=0 equation

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The solution

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 5         
z  + 32 = 0
$$z^{5} + 32 = 0$$
Detail solution
Given the equation
$$z^{5} + 32 = 0$$
Because equation degree is equal to = 5 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 5-th degree of the equation sides:
We get:
$$\sqrt[5]{z^{5}} = \sqrt[5]{-32}$$
or
$$z = 2 \sqrt[5]{-1}$$
Expand brackets in the right part
z = -2*1^1/5

We get the answer: z = 2*(-1)^(1/5)

All other 4 root(s) is the complex numbers.
do replacement:
$$w = z$$
then the equation will be the:
$$w^{5} = -32$$
Any complex number can presented so:
$$w = r e^{i p}$$
substitute to the equation
$$r^{5} e^{5 i p} = -32$$
where
$$r = 2$$
- the magnitude of the complex number
Substitute r:
$$e^{5 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = -1$$
so
$$\cos{\left(5 p \right)} = -1$$
and
$$\sin{\left(5 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{5} + \frac{\pi}{5}$$
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:
$$w_{1} = -2$$
$$w_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$w_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$w_{4} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$w_{5} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
do backward replacement
$$w = z$$
$$z = w$$

The final answer:
$$z_{1} = -2$$
$$z_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{4} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$z_{5} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
Sum and product of roots [src]
sum
                                                                         /       ___________        ___________              ___________              ___________\                                               /     ___________        ___________              ___________              ___________\                                                                         
                                                                         |      /       ___        /       ___              /       ___              /       ___ |                                               |    /       ___        /       ___              /       ___              /       ___ |                                                                         
                   /       ___________              ___________\         |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________         |   /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________                        ___________
           ___     |      /       ___              /       ___ |         |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___          |  /   - + -----      /   - - -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___          ___           /       ___ 
     1   \/ 5      |     /  5   \/ 5       ___    /  5   \/ 5  |   1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5     1     |\/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5     1   \/ 5           /  5   \/ 5  
-2 + - - ----- + I*|-   /   - - -----  - \/ 5 *  /   - - ----- | + - + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*  /   - - ----- *  /   - + -----  + - + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2*  /   - - ----- *  /   - + -----  + - + ----- + 2*I*  /   - - ----- 
     2     2       \  \/    8     8            \/    8     8   /   2     \         2                  2                     2                        2           /     \/    8     8    \/    8     8      2     \       2                  2                     2                        2           /     \/    8     8    \/    8     8      2     2         \/    8     8   
$$\left(\left(\left(-2 + \left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right)\right) + \left(\frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right)\right) + \left(- 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
=
                                                    /     ___________        ___________              ___________              ___________\     /       ___________        ___________              ___________              ___________\                       
                                                    |    /       ___        /       ___              /       ___              /       ___ |     |      /       ___        /       ___              /       ___              /       ___ |                       
  /       ___________              ___________\     |   /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |     |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |            ___________
  |      /       ___              /       ___ |     |  /   - + -----      /   - - -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |     |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |           /       ___ 
  |     /  5   \/ 5       ___    /  5   \/ 5  |     |\/    8     8      \/    8     8            \/    8     8            \/    8     8   |     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |          /  5   \/ 5  
I*|-   /   - - -----  - \/ 5 *  /   - - ----- | + I*|---------------- - ---------------- + ---------------------- + ----------------------| + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*I*  /   - - ----- 
  \  \/    8     8            \/    8     8   /     \       2                  2                     2                        2           /     \         2                  2                     2                        2           /       \/    8     8   
$$i \left(- \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)$$
product
                                                                 /      /       ___________        ___________              ___________              ___________\                                      \ /      /     ___________        ___________              ___________              ___________\                                      \                                   
                                                                 |      |      /       ___        /       ___              /       ___              /       ___ |                                      | |      |    /       ___        /       ___              /       ___              /       ___ |                                      |                                   
   /              /       ___________              ___________\\ |      |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________| |      |   /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________| /                     ___________\
   |      ___     |      /       ___              /       ___ || |      |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___ | |      |  /   - + -----      /   - - -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___ | |      ___           /       ___ |
   |1   \/ 5      |     /  5   \/ 5       ___    /  5   \/ 5  || |1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5  | |1     |\/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5  | |1   \/ 5           /  5   \/ 5  |
-2*|- - ----- + I*|-   /   - - -----  - \/ 5 *  /   - - ----- ||*|- + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*  /   - - ----- *  /   - + ----- |*|- + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2*  /   - - ----- *  /   - + ----- |*|- + ----- + 2*I*  /   - - ----- |
   \2     2       \  \/    8     8            \/    8     8   // \2     \         2                  2                     2                        2           /     \/    8     8    \/    8     8   / \2     \       2                  2                     2                        2           /     \/    8     8    \/    8     8   / \2     2         \/    8     8   /
$$\left(\frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right) \left(- 2 \left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right)\right) \left(- 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right) \left(\frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
=
                                                                                         ___________                                          ___________          ____________    ___________                ____________    ___________
                           ____________        ____________    ___________       ____   /       ___                ___________         ___   /       ___          /        ___    /       ___          ___   /        ___    /       ___ 
          ___       ___   /        ___        /        ___    /       ___    I*\/ 10 *\/  5 - \/ 5         ____   /       ___    3*I*\/ 2 *\/  5 - \/ 5     5*I*\/  -5 + \/ 5  *\/  5 + \/ 5     3*I*\/ 5 *\/  -5 + \/ 5  *\/  5 + \/ 5  
-12 + 4*\/ 5  + 4*\/ 2 *\/  -5 + \/ 5   + I*\/  -5 + \/ 5  *\/  5 - \/ 5   + ----------------------- - I*\/ 10 *\/  5 + \/ 5   - ------------------------ + ---------------------------------- + ----------------------------------------
                                                                                        2                                                   2                               2                                       2                    
$$\frac{3 \sqrt{5} i \sqrt{-5 + \sqrt{5}} \sqrt{\sqrt{5} + 5}}{2} - 12 + \frac{5 i \sqrt{-5 + \sqrt{5}} \sqrt{\sqrt{5} + 5}}{2} + i \sqrt{-5 + \sqrt{5}} \sqrt{5 - \sqrt{5}} + 4 \sqrt{5} - \sqrt{10} i \sqrt{\sqrt{5} + 5} - \frac{3 \sqrt{2} i \sqrt{5 - \sqrt{5}}}{2} + \frac{\sqrt{10} i \sqrt{5 - \sqrt{5}}}{2} + 4 \sqrt{2} \sqrt{-5 + \sqrt{5}}$$
-12 + 4*sqrt(5) + 4*sqrt(2)*sqrt(-5 + sqrt(5)) + i*sqrt(-5 + sqrt(5))*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5))/2 - i*sqrt(10)*sqrt(5 + sqrt(5)) - 3*i*sqrt(2)*sqrt(5 - sqrt(5))/2 + 5*i*sqrt(-5 + sqrt(5))*sqrt(5 + sqrt(5))/2 + 3*i*sqrt(5)*sqrt(-5 + sqrt(5))*sqrt(5 + sqrt(5))/2
Rapid solution [src]
z1 = -2
$$z_{1} = -2$$
                   /       ___________              ___________\
           ___     |      /       ___              /       ___ |
     1   \/ 5      |     /  5   \/ 5       ___    /  5   \/ 5  |
z2 = - - ----- + I*|-   /   - - -----  - \/ 5 *  /   - - ----- |
     2     2       \  \/    8     8            \/    8     8   /
$$z_{2} = - \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
           /       ___________        ___________              ___________              ___________\                                      
           |      /       ___        /       ___              /       ___              /       ___ |                                      
           |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________
           |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___ 
     1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5  
z3 = - + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*  /   - - ----- *  /   - + ----- 
     2     \         2                  2                     2                        2           /     \/    8     8    \/    8     8   
$$z_{3} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)$$
           /     ___________        ___________              ___________              ___________\                                      
           |    /       ___        /       ___              /       ___              /       ___ |                                      
           |   /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |          ___________      ___________
           |  /   - + -----      /   - - -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |         /       ___      /       ___ 
     1     |\/    8     8      \/    8     8            \/    8     8            \/    8     8   |        /  5   \/ 5      /  5   \/ 5  
z4 = - + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2*  /   - - ----- *  /   - + ----- 
     2     \       2                  2                     2                        2           /     \/    8     8    \/    8     8   
$$z_{4} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)$$
                          ___________
           ___           /       ___ 
     1   \/ 5           /  5   \/ 5  
z5 = - + ----- + 2*I*  /   - - ----- 
     2     2         \/    8     8   
$$z_{5} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
z5 = 1/2 + sqrt(5)/2 + 2*i*sqrt(5/8 - sqrt(5)/8)
Numerical answer [src]
z1 = 1.61803398874989 + 1.17557050458495*i
z2 = -2.0
z3 = 1.61803398874989 - 1.17557050458495*i
z4 = -0.618033988749895 - 1.90211303259031*i
z5 = -0.618033988749895 + 1.90211303259031*i
z5 = -0.618033988749895 + 1.90211303259031*i
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