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x^4=625

x^4=625 equation

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Numerical solution:

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The solution

You have entered [src] 
 4      
x  = 625
x4=625x^{4} = 625
Simplify
Detail solution
Given the equation
x4=625x^{4} = 625
Because equation degree is equal to = 4 - contains the even number 4 in the numerator, then
the equation has two real roots.
Get the root 4-th degree of the equation sides:
We get:
x44=6254\sqrt[4]{x^{4}} = \sqrt[4]{625}
x44=(−1)6254\sqrt[4]{x^{4}} = \left(-1\right) \sqrt[4]{625}
or
x=5x = 5
x=−5x = -5
We get the answer: x = 5
We get the answer: x = -5
or
x1=−5x_{1} = -5
x2=5x_{2} = 5

All other 2 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
z4=625z^{4} = 625
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r4e4ip=625r^{4} e^{4 i p} = 625
where
r=5r = 5
- the magnitude of the complex number
Substitute r:
e4ip=1e^{4 i p} = 1
Using Euler’s formula, we find roots for p
isin⁡(4p)+cos⁡(4p)=1i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = 1
so
cos⁡(4p)=1\cos{\left(4 p \right)} = 1
and
sin⁡(4p)=0\sin{\left(4 p \right)} = 0
then
p=πN2p = \frac{\pi N}{2}
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=−5z_{1} = -5
z2=5z_{2} = 5
z3=−5iz_{3} = - 5 i
z4=5iz_{4} = 5 i
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−5x_{1} = -5
x2=5x_{2} = 5
x3=−5ix_{3} = - 5 i
x4=5ix_{4} = 5 i
The graph
05-20-15-10-51015200100000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -5
x1=−5x_{1} = -5 
x2 = 5
x2=5x_{2} = 5 
x3 = -5*I
x3=−5ix_{3} = - 5 i 
x4 = 5*I
x4=5ix_{4} = 5 i 
x4 = 5*i
Sum and product of roots [src] 
sum
-5 + 5 - 5*I + 5*I
((−5+5)−5i)+5i\left(\left(-5 + 5\right) - 5 i\right) + 5 i 
=
0
00 
product
-5*5*-5*I*5*I
5i−25(−5i)5 i - 25 \left(- 5 i\right) 
=
-625
−625-625 
-625
Numerical answer [src] 
x1 = 5.0*i
x2 = 5.0
x3 = -5.0*i
x4 = -5.0
x4 = -5.0
The graph
x^4=625 equation
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