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z^5=-1

z^5=-1 equation

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Numerical solution:

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The solution

You have entered [src] 
 5     
z  = -1
z5=−1z^{5} = -1
Simplify
Detail solution
Given the equation
z5=−1z^{5} = -1
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:
(1z+0)55=−15\sqrt[5]{\left(1 z + 0\right)^{5}} = \sqrt[5]{-1}
or
z=−15z = \sqrt[5]{-1}
Expand brackets in the right part
z = -1^1/5

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

All other 4 root(s) is the complex numbers.
do replacement:
w=zw = z
then the equation will be the:
w5=−1w^{5} = -1
Any complex number can presented so:
w=reipw = r e^{i p}
substitute to the equation
r5e5ip=−1r^{5} e^{5 i p} = -1
where
r=1r = 1
- the magnitude of the complex number
Substitute r:
e5ip=−1e^{5 i p} = -1
Using Euler’s formula, we find roots for p
isin⁡(5p)+cos⁡(5p)=−1i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = -1
so
cos⁡(5p)=−1\cos{\left(5 p \right)} = -1
and
sin⁡(5p)=0\sin{\left(5 p \right)} = 0
then
p=2πN5+π5p = \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:
w1=−1w_{1} = -1
w2=14+54+i−58+58w_{2} = \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}
w3=−54+14−5i−58+582−i−58+582w_{3} = - \frac{\sqrt{5}}{4} + \frac{1}{4} - \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}
w4=14+−58+5858+58−5i58+584−i58+584−i−58+584+5i−58+584w_{4} = \frac{1}{4} + \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}
w5=−−58+5858+58+14−i−58+584+i58+584+5i−58+584+5i58+584w_{5} = - \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}
do backward replacement
w=zw = z
z=wz = w

The final answer:
z1=−1z_{1} = -1
z2=14+54+i−58+58z_{2} = \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}
z3=−54+14−5i−58+582−i−58+582z_{3} = - \frac{\sqrt{5}}{4} + \frac{1}{4} - \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}
z4=14+−58+5858+58−5i58+584−i58+584−i−58+584+5i−58+584z_{4} = \frac{1}{4} + \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}
z5=−−58+5858+58+14−i−58+584+i58+584+5i−58+584+5i58+584z_{5} = - \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}
The graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-200000200000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
z1 = -1
z1=−1z_{1} = -1
Simplify 
                   /       ___________              ___________\
                   |      /       ___              /       ___ |
                   |     /  5   \/ 5       ___    /  5   \/ 5  |
           ___     |    /   - - -----    \/ 5 *  /   - - ----- |
     1   \/ 5      |  \/    8     8            \/    8     8   |
z2 = - - ----- + I*|- ---------------- - ----------------------|
     4     4       \         2                     2           /
z2=−54+14+i(−5−58+582−−58+582)z_{2} = - \frac{\sqrt{5}}{4} + \frac{1}{4} + i \left(- \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)
Simplify 
           /       ___________        ___________              ___________              ___________\                                    
           |      /       ___        /       ___              /       ___              /       ___ |                                    
           |     /  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*|- ---------------- - ---------------- - ---------------------- + ----------------------| +   /   - - ----- *  /   - + ----- 
     4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8
z3=14+−58+5858+58+i(−558+584−58+584−−58+584+5−58+584)z_{3} = \frac{1}{4} + \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)
Simplify 
           /       ___________        ___________              ___________              ___________\                                    
           |      /       ___        /       ___              /       ___              /       ___ |                                    
           |     /  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*|- ---------------- + ---------------- + ---------------------- + ----------------------| -   /   - - ----- *  /   - + ----- 
     4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8
z4=−−58+5858+58+14+i(−−58+584+58+584+5−58+584+558+584)z_{4} = - \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} + i \left(- \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)
Simplify 
                        ___________
           ___         /       ___ 
     1   \/ 5         /  5   \/ 5  
z5 = - + ----- + I*  /   - - ----- 
     4     4       \/    8     8
z5=14+54+i−58+58z_{5} = \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}
Simplify
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  |        ___________      ___________                      ___________
              ___     |    /   - - -----    \/ 5 *  /   - - ----- |         |    /   - - -----      /   - + -----    \/ 5 *  /   - + -----    \/ 5 *  /   - - ----- |       /       ___      /       ___          |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |       /       ___      /       ___          ___         /       ___ 
        1   \/ 5      |  \/    8     8            \/    8     8   |   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  
0 - 1 + - - ----- + I*|- ---------------- - ----------------------| + - + I*|- ---------------- - ---------------- - ---------------------- + ----------------------| +   /   - - ----- *  /   - + -----  + - + I*|- ---------------- + ---------------- + ---------------------- + ----------------------| -   /   - - ----- *  /   - + -----  + - + ----- + I*  /   - - ----- 
        4     4       \         2                     2           /   4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8      4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8      4     4       \/    8     8
((((−1+0)+(−54+14+i(−5−58+582−−58+582)))+(14+−58+5858+58+i(−558+584−58+584−−58+584+5−58+584)))+(−−58+5858+58+14+i(−−58+584+58+584+5−58+584+558+584)))+(14+54+i−58+58)\left(\left(\left(\left(-1 + 0\right) + \left(- \frac{\sqrt{5}}{4} + \frac{1}{4} + i \left(- \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right)\right) + \left(\frac{1}{4} + \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)\right)\right) + \left(- \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} + i \left(- \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)\right)\right) + \left(\frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{- \frac{\sqrt{5}}{8} + \frac{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 *  /   - - -----    \/ 5 *  /   - + ----- |
     /  5   \/ 5       |  \/    8     8            \/    8     8   |     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |
I*  /   - - -----  + I*|- ---------------- - ----------------------| + I*|- ---------------- - ---------------- - ---------------------- + ----------------------| + I*|- ---------------- + ---------------- + ---------------------- + ----------------------|
  \/    8     8        \         2                     2           /     \         4                  4                     4                        4           /     \         4                  4                     4                        4           /
i(−5−58+582−−58+582)+i(−558+584−58+584−−58+584+5−58+584)+i−58+58+i(−−58+584+58+584+5−58+584+558+584)i \left(- \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right) + i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right) 
product
     /              /       ___________              ___________\\ /      /       ___________        ___________              ___________              ___________\                                    \ /      /       ___________        ___________              ___________              ___________\                                    \                                 
     |              |      /       ___              /       ___ || |      |      /       ___        /       ___              /       ___              /       ___ |                                    | |      |      /       ___        /       ___              /       ___              /       ___ |                                    |                                 
     |              |     /  5   \/ 5       ___    /  5   \/ 5  || |      |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |        ___________      ___________| |      |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |        ___________      ___________| /                   ___________\
     |      ___     |    /   - - -----    \/ 5 *  /   - - ----- || |      |    /   - - -----      /   - + -----    \/ 5 *  /   - + -----    \/ 5 *  /   - - ----- |       /       ___      /       ___ | |      |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |       /       ___      /       ___ | |      ___         /       ___ |
     |1   \/ 5      |  \/    8     8            \/    8     8   || |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  |
1*-1*|- - ----- + I*|- ---------------- - ----------------------||*|- + I*|- ---------------- - ---------------- - ---------------------- + ----------------------| +   /   - - ----- *  /   - + ----- |*|- + I*|- ---------------- + ---------------- + ---------------------- + ----------------------| -   /   - - ----- *  /   - + ----- |*|- + ----- + I*  /   - - ----- |
     \4     4       \         2                     2           // \4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8   / \4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8   / \4     4       \/    8     8   /
1(−1)(−54+14+i(−5−58+582−−58+582))(14+−58+5858+58+i(−558+584−58+584−−58+584+5−58+584))(−−58+5858+58+14+i(−−58+584+58+584+5−58+584+558+584))(14+54+i−58+58)1 \left(-1\right) \left(- \frac{\sqrt{5}}{4} + \frac{1}{4} + i \left(- \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right) \left(\frac{1}{4} + \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)\right) \left(- \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} + i \left(- \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)\right) \left(\frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}\right) 
=
          _______________        _______________          ______________
         /           ___        /           ___          /          ___ 
     I*\/  50 + 10*\/ 5     I*\/  50 - 10*\/ 5     5*I*\/  10 - 2*\/ 5  
-1 - -------------------- + -------------------- + ---------------------
              32                     64                      64
−1−i105+5032+i−105+5064+5i−25+1064-1 - \frac{i \sqrt{10 \sqrt{5} + 50}}{32} + \frac{i \sqrt{- 10 \sqrt{5} + 50}}{64} + \frac{5 i \sqrt{- 2 \sqrt{5} + 10}}{64} 
-1 - i*sqrt(50 + 10*sqrt(5))/32 + i*sqrt(50 - 10*sqrt(5))/64 + 5*i*sqrt(10 - 2*sqrt(5))/64
Numerical answer [src] 
z1 = 0.809016994374947 + 0.587785252292473*i
z2 = 0.809016994374947 - 0.587785252292473*i
z3 = -0.309016994374947 + 0.951056516295154*i
z4 = -0.309016994374947 - 0.951056516295154*i
z5 = -1.0
z5 = -1.0
The graph
z^5=-1 equation
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