MATLAB: Why does similar code generate complex numbers

complex numbersMATLAB

In the following code, the array 'f1' contains real numbers while the array 'f2' contains complex numbers. From what I understand and am expecting, both arrays should contain identical data, consisting only of real numbers. But this is not what I get. Instead, the array 'f2', generated using exponential notation, contains complex numbers and the question is why is this happening?
x = [-27 -8 -1 1 8 27];
f1   = nthroot(x, 3);
f2   = x .^ (1/3);
The array 'f1' contains the expected results:
-3 -2 -1 1 2 3
For the array 'f2', here are the numbers generated:
1.5000 + 2.5981i 1.0000 + 1.7321i 0.5000 + 0.8660i 1.0000 + 0.0000i 2.0000 + 0.0000i 3.0000 + 0.0000i
The array 'f2' unexpectedly contains complex numbers. The last 3 numbers give the correct/expected result but the first 3 numbers are not the correct/expected result. That is, the first 3 numbers in 'f2' are NOT the cubed root of the first 3 numbers in 'x'. Why is this happening?
In trying to debug/understand the problem, when I manually enter the negative numbers of 'x' in the command window then I get the expected results:
-27 ^ (1/3) —> ans = -3
-8 ^ (1/3) —> ans = -2
-1 ^ (1/3) —> ans = -1
However, when I use array syntax in the command window and select the same negative numbers from 'x' then I get unexpected and incorrect results:
x(1) ^ (1/3) —> ans = 1.5000 + 2.5981i
x(2) ^ (1/3) —> ans = 1.0000 + 1.7321i
x(3) ^ (1/3) —> ans = 0.5000 + 0.8660i
I have no experience working with complex numbers. The question I am asking is why do I get incorrect results (i.e. in complex number format) when using exponential notation in my script?
Thanks to all who respond

Best Answer

When x is negative, x^n is defined by MATLAB as exp(n*log(x)) . log(x) when x is negative, is log(-x) + pi * 1i . n times that is n*log(-x) + n*pi*1i . exp() of that is going to be complex except when n is an integer (in which case n*pi becomes an exact multiple of pi)
Thus for x negative, f2 = x .^ (1/3); is going to be exp(log(-x)/3 + pi/3*1i) which is going to be complex valued.
nthroot() is defined in terms of real roots, with it being an error for the value to be negative and the power to not be an odd integer.
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