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x^8-1=0

x^8-1=0 equation

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Numerical solution:

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The solution

You have entered [src] 
 8        
x  - 1 = 0
x8−1=0x^{8} - 1 = 0
Simplify
Detail solution
Given the equation
x8−1=0x^{8} - 1 = 0
Because equation degree is equal to = 8 - contains the even number 8 in the numerator, then
the equation has two real roots.
Get the root 8-th degree of the equation sides:
We get:
x88=18\sqrt[8]{x^{8}} = \sqrt[8]{1}
x88=(−1)18\sqrt[8]{x^{8}} = \left(-1\right) \sqrt[8]{1}
or
x=1x = 1
x=−1x = -1
We get the answer: x = 1
We get the answer: x = -1
or
x1=−1x_{1} = -1
x2=1x_{2} = 1

All other 6 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
z8=1z^{8} = 1
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r8e8ip=1r^{8} e^{8 i p} = 1
where
r=1r = 1
- the magnitude of the complex number
Substitute r:
e8ip=1e^{8 i p} = 1
Using Euler’s formula, we find roots for p
isin⁡(8p)+cos⁡(8p)=1i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = 1
so
cos⁡(8p)=1\cos{\left(8 p \right)} = 1
and
sin⁡(8p)=0\sin{\left(8 p \right)} = 0
then
p=πN4p = \frac{\pi N}{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:
z1=−1z_{1} = -1
z2=1z_{2} = 1
z3=−iz_{3} = - i
z4=iz_{4} = i
z5=−22−2i2z_{5} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}
z6=−22+2i2z_{6} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}
z7=22−2i2z_{7} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}
z8=22+2i2z_{8} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−1x_{1} = -1
x2=1x_{2} = 1
x3=−ix_{3} = - i
x4=ix_{4} = i
x5=−22−2i2x_{5} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}
x6=−22+2i2x_{6} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}
x7=22−2i2x_{7} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}
x8=22+2i2x_{8} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}
The graph
05-15-10-51015-250000000250000000
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
                     ___       ___       ___       ___     ___       ___     ___       ___
                   \/ 2    I*\/ 2      \/ 2    I*\/ 2    \/ 2    I*\/ 2    \/ 2    I*\/ 2 
-1 + 1 - I + I + - ----- - ------- + - ----- + ------- + ----- - ------- + ----- + -------
                     2        2          2        2        2        2        2        2
((22−2i2)+(((−22−2i2)+(((−1+1)−i)+i))+(−22+2i2)))+(22+2i2)\left(\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(\left(\left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(\left(\left(-1 + 1\right) - i\right) + i\right)\right) + \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)\right)\right) + \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) 
=
0
00 
product
        /    ___       ___\ /    ___       ___\ /  ___       ___\ /  ___       ___\
        |  \/ 2    I*\/ 2 | |  \/ 2    I*\/ 2 | |\/ 2    I*\/ 2 | |\/ 2    I*\/ 2 |
-(-I)*I*|- ----- - -------|*|- ----- + -------|*|----- - -------|*|----- + -------|
        \    2        2   / \    2        2   / \  2        2   / \  2        2   /
i(−(−1)i)(−22−2i2)(−22+2i2)(22−2i2)(22+2i2)i \left(- \left(-1\right) i\right) \left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) 
=
-1
−1-1 
-1
Rapid solution [src] 
x1 = -1
x1=−1x_{1} = -1 
x2 = 1
x2=1x_{2} = 1 
x3 = -I
x3=−ix_{3} = - i 
x4 = I
x4=ix_{4} = i 
         ___       ___
       \/ 2    I*\/ 2 
x5 = - ----- - -------
         2        2
x5=−22−2i2x_{5} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} 
         ___       ___
       \/ 2    I*\/ 2 
x6 = - ----- + -------
         2        2
x6=−22+2i2x_{6} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} 
       ___       ___
     \/ 2    I*\/ 2 
x7 = ----- - -------
       2        2
x7=22−2i2x_{7} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} 
       ___       ___
     \/ 2    I*\/ 2 
x8 = ----- + -------
       2        2
x8=22+2i2x_{8} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} 
x8 = sqrt(2)/2 + sqrt(2)*i/2
Numerical answer [src] 
x1 = -0.707106781186548 - 0.707106781186548*i
x2 = 0.707106781186548 + 0.707106781186548*i
x3 = -0.707106781186548 + 0.707106781186548*i
x4 = 1.0*i
x5 = -1.0
x6 = 0.707106781186548 - 0.707106781186548*i
x7 = 1.0
x8 = -1.0*i
x8 = -1.0*i
The graph
x^8-1=0 equation
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