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x^6=5

x^6=5 equation

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Numerical solution:

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The solution

You have entered [src] 
 6    
x  = 5
x6=5x^{6} = 5
Simplify
Detail solution
Given the equation
x6=5x^{6} = 5
Because equation degree is equal to = 6 - contains the even number 6 in the numerator, then
the equation has two real roots.
Get the root 6-th degree of the equation sides:
We get:
(1x+0)66=56\sqrt[6]{\left(1 x + 0\right)^{6}} = \sqrt[6]{5}
(1x+0)66=−56\sqrt[6]{\left(1 x + 0\right)^{6}} = - \sqrt[6]{5}
or
x=56x = \sqrt[6]{5}
x=−56x = - \sqrt[6]{5}
Expand brackets in the right part
x = 5^1/6

We get the answer: x = 5^(1/6)
Expand brackets in the right part
x = -5^1/6

We get the answer: x = -5^(1/6)
or
x1=−56x_{1} = - \sqrt[6]{5}
x2=56x_{2} = \sqrt[6]{5}

All other 4 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
z6=5z^{6} = 5
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r6e6ip=5r^{6} e^{6 i p} = 5
where
r=56r = \sqrt[6]{5}
- the magnitude of the complex number
Substitute r:
e6ip=1e^{6 i p} = 1
Using Euler’s formula, we find roots for p
isin⁡(6p)+cos⁡(6p)=1i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = 1
so
cos⁡(6p)=1\cos{\left(6 p \right)} = 1
and
sin⁡(6p)=0\sin{\left(6 p \right)} = 0
then
p=πN3p = \frac{\pi N}{3}
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=−56z_{1} = - \sqrt[6]{5}
z2=56z_{2} = \sqrt[6]{5}
z3=−562−3⋅56i2z_{3} = - \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
z4=−562+3⋅56i2z_{4} = - \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
z5=562−3⋅56i2z_{5} = \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
z6=562+3⋅56i2z_{6} = \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−56x_{1} = - \sqrt[6]{5}
x2=56x_{2} = \sqrt[6]{5}
x3=−562−3⋅56i2x_{3} = - \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
x4=−562+3⋅56i2x_{4} = - \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
x5=562−3⋅56i2x_{5} = \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
x6=562+3⋅56i2x_{6} = \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
The graph
05-15-10-5101504000000
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
                      6 ___       ___ 6 ___     6 ___       ___ 6 ___   6 ___       ___ 6 ___   6 ___       ___ 6 ___
    6 ___   6 ___     \/ 5    I*\/ 3 *\/ 5      \/ 5    I*\/ 3 *\/ 5    \/ 5    I*\/ 3 *\/ 5    \/ 5    I*\/ 3 *\/ 5 
0 - \/ 5  + \/ 5  + - ----- - ------------- + - ----- + ------------- + ----- - ------------- + ----- + -------------
                        2           2             2           2           2           2           2           2
((562−3⋅56i2)+((((−56+0)+56)−(562+3⋅56i2))−(562−3⋅56i2)))+(562+3⋅56i2)\left(\left(\frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) + \left(\left(\left(\left(- \sqrt[6]{5} + 0\right) + \sqrt[6]{5}\right) - \left(\frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right)\right) - \left(\frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right)\right)\right) + \left(\frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) 
=
0
00 
product
               /  6 ___       ___ 6 ___\ /  6 ___       ___ 6 ___\ /6 ___       ___ 6 ___\ /6 ___       ___ 6 ___\
   6 ___ 6 ___ |  \/ 5    I*\/ 3 *\/ 5 | |  \/ 5    I*\/ 3 *\/ 5 | |\/ 5    I*\/ 3 *\/ 5 | |\/ 5    I*\/ 3 *\/ 5 |
1*-\/ 5 *\/ 5 *|- ----- - -------------|*|- ----- + -------------|*|----- - -------------|*|----- + -------------|
               \    2           2      / \    2           2      / \  2           2      / \  2           2      /
56⋅1(−56)(−562−3⋅56i2)(−562+3⋅56i2)(562−3⋅56i2)(562+3⋅56i2)\sqrt[6]{5} \cdot 1 \left(- \sqrt[6]{5}\right) \left(- \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) \left(- \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) \left(\frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) \left(\frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}\right) 
=
-5
−5-5 
-5
Rapid solution [src] 
      6 ___
x1 = -\/ 5
x1=−56x_{1} = - \sqrt[6]{5}
Simplify 
     6 ___
x2 = \/ 5
x2=56x_{2} = \sqrt[6]{5}
Simplify 
       6 ___       ___ 6 ___
       \/ 5    I*\/ 3 *\/ 5 
x3 = - ----- - -------------
         2           2
x3=−562−3⋅56i2x_{3} = - \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
Simplify 
       6 ___       ___ 6 ___
       \/ 5    I*\/ 3 *\/ 5 
x4 = - ----- + -------------
         2           2
x4=−562+3⋅56i2x_{4} = - \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
Simplify 
     6 ___       ___ 6 ___
     \/ 5    I*\/ 3 *\/ 5 
x5 = ----- - -------------
       2           2
x5=562−3⋅56i2x_{5} = \frac{\sqrt[6]{5}}{2} - \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
Simplify 
     6 ___       ___ 6 ___
     \/ 5    I*\/ 3 *\/ 5 
x6 = ----- + -------------
       2           2
x6=562+3⋅56i2x_{6} = \frac{\sqrt[6]{5}}{2} + \frac{\sqrt{3} \cdot \sqrt[6]{5} i}{2}
Simplify
Numerical answer [src] 
x1 = -0.653830243005915 - 1.13246720041135*i
x2 = 0.653830243005915 + 1.13246720041135*i
x3 = -1.30766048601183
x4 = 1.30766048601183
x5 = 0.653830243005915 - 1.13246720041135*i
x6 = -0.653830243005915 + 1.13246720041135*i
x6 = -0.653830243005915 + 1.13246720041135*i
The graph
x^6=5 equation
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