To find the value of z, we can start by finding the real and imaginary parts of z.

Given z = 5 - 4i, we have:
Real part = 5
Imaginary part = -4

Now, let's find the magnitude of z using the Pythagorean theorem:
|z| = √(5^2 + (-4)^2)
|z| = √(25 + 16)
|z| = √41

Therefore, the magnitude of z is √41.

Next, let's find the argument of z using the inverse tangent function:
Arg(z) = arctan(-4/5)
Arg(z) ≈ -0.6747 radians

So, the polar form of z is z = √41(cos(-0.6747) + isin(-0.6747)).

Finally, to find z^1 = 4 - i, we can express this number in polar form as well:
z^1 = √(4^2 + (-1)^2)(cos(arctan(-1/4)) + isin(arctan(-1/4)))
z^1 = √17(cos(-0.2449) + isin(-0.2449))

Therefore, z = √41(cos(-0.6747) + isin(-0.6747)) and z^1 = √17(cos(-0.2449) + isin(-0.2449).