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x^6=64

x^6=64 equation

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Numerical solution:

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The solution

You have entered [src] 
 6     
x  = 64
x6=64x^{6} = 64
Simplify
Detail solution
Given the equation
x6=64x^{6} = 64
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=2\sqrt[6]{\left(1 x + 0\right)^{6}} = 2
(1x+0)66=−2\sqrt[6]{\left(1 x + 0\right)^{6}} = -2
or
x=2x = 2
x=−2x = -2
We get the answer: x = 2
We get the answer: x = -2
or
x1=−2x_{1} = -2
x2=2x_{2} = 2

All other 4 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
z6=64z^{6} = 64
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r6e6ip=64r^{6} e^{6 i p} = 64
where
r=2r = 2
- 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=−2z_{1} = -2
z2=2z_{2} = 2
z3=−1−3iz_{3} = -1 - \sqrt{3} i
z4=−1+3iz_{4} = -1 + \sqrt{3} i
z5=1−3iz_{5} = 1 - \sqrt{3} i
z6=1+3iz_{6} = 1 + \sqrt{3} i
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−2x_{1} = -2
x2=2x_{2} = 2
x3=−1−3ix_{3} = -1 - \sqrt{3} i
x4=−1+3ix_{4} = -1 + \sqrt{3} i
x5=1−3ix_{5} = 1 - \sqrt{3} i
x6=1+3ix_{6} = 1 + \sqrt{3} i
The graph
05-15-10-5101505000000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -2
x1=−2x_{1} = -2
Simplify 
x2 = 2
x2=2x_{2} = 2
Simplify 
              ___
x3 = -1 - I*\/ 3
x3=−1−3ix_{3} = -1 - \sqrt{3} i
Simplify 
              ___
x4 = -1 + I*\/ 3
x4=−1+3ix_{4} = -1 + \sqrt{3} i
Simplify 
             ___
x5 = 1 - I*\/ 3
x5=1−3ix_{5} = 1 - \sqrt{3} i
Simplify 
             ___
x6 = 1 + I*\/ 3
x6=1+3ix_{6} = 1 + \sqrt{3} i
Simplify
Sum and product of roots [src] 
sum
                     ___            ___           ___           ___
0 - 2 + 2 + -1 - I*\/ 3  + -1 + I*\/ 3  + 1 - I*\/ 3  + 1 + I*\/ 3
((1−3i)+((((−2+0)+2)−(1+3i))−(1−3i)))+(1+3i)\left(\left(1 - \sqrt{3} i\right) + \left(\left(\left(\left(-2 + 0\right) + 2\right) - \left(1 + \sqrt{3} i\right)\right) - \left(1 - \sqrt{3} i\right)\right)\right) + \left(1 + \sqrt{3} i\right) 
=
0
00 
product
       /         ___\ /         ___\ /        ___\ /        ___\
1*-2*2*\-1 - I*\/ 3 /*\-1 + I*\/ 3 /*\1 - I*\/ 3 /*\1 + I*\/ 3 /
1(−2)2(−1−3i)(−1+3i)(1−3i)(1+3i)1 \left(-2\right) 2 \left(-1 - \sqrt{3} i\right) \left(-1 + \sqrt{3} i\right) \left(1 - \sqrt{3} i\right) \left(1 + \sqrt{3} i\right) 
=
-64
−64-64 
-64
Numerical answer [src] 
x1 = -1.0 - 1.73205080756888*i
x2 = 1.0 + 1.73205080756888*i
x3 = -1.0 + 1.73205080756888*i
x4 = 2.0
x5 = 1.0 - 1.73205080756888*i
x6 = -2.0
x6 = -2.0
The graph
x^6=64 equation
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