A function is even if f(-x) = f(x), while it is odd if f(-x) = -1*f(x)... These equations have to be true for all x.
If f(-x) does not equal either f(x) or -1*f(x), then it can't be called either even or odd.
So, let's evaluate f(-x):
(-x)^3 - 7*(-x) = (-1)^3*(x)^3...

What if the question had been [(a^2)*(b^3)*c]^4?
I think the operating rule is how to deal with a power raised to another power, instead of the multiplication of the same base raised to two different powers. That says
(x^y)^z = x^(y*z)
...or we multiply the two exponents...

You have a multiplication, raised to a power.
So, there's a theory that says:
(a*b)^c = (a^c)*(b^c)
- this can be proven by expanding and rearranging the order of the multiplications. As an example:
(2*3)^4 = (2*3)*(2*3)*(2*3)*(2*3) = (2*2*2*2)*(3*3*3*3) = (2^4)*(3^4)...