The polar form of a complex number is Re^(itheta), where R is the number's modulus (its distance from 0) and theta is the angle formed by the positive real axis and the number's vector on the complex plane.
We have a nice way of converting to polar coordinates by using Euler's formula: e^(itheta) = cos(theta)+isin(theta). Thus, if we can find and factor out R, we can find (theta) from the remaining number.
In this case, we will first find 2+i in polar form, and then apply the power of 1/2.
To find R, we find the number's modulus: |a+bi| = sqrt(a^2+b^2)
|2+i| = sqrt(2^2+1^2) = sqrt(5)
=> 2+i = sqrt(5)(2/sqrt(5)+1/sqrt(5)i)
So, we have cos(theta) = 2/sqrt(5) and sin(theta)=1/sqrt(5)
As arccos(2/sqrt(5))=arcsin(1/sqrt(5))~~0.4636 is not one of the "nice" angles, we'll leave it in that form. For ease of use, let's let theta_0 = arccos(2/sqrt(5)) and write that for the remainder of the problem.
Proceeding, we now have
2+i = sqrt(5)(cos(theta_0)+isin(theta_0))
By Euler's formula, this gives us
2+i = sqrt(5)e^(itheta_0)
Note that we can add 2pii in any integer multiple without changing the value due to the cyclic nature of sin(theta) and cos(theta). This will become relevant once we take the root.
2+i = sqrt(5)e^(i(theta_0+2pin))
Finally, we take a power of 1/2 to get
sqrt(2+i) = (sqrt(5)e^(itheta_0))^(1/2)
=5^(1/4)e^(i(theta_0+2pin)/2)
=5^(1/4)e^(i(theta_0/2+pin))
