Answer
Odd positive integer value of $m$
Work Step by Step
If $m$ is an even positive integer we have:
$$\sqrt{2^m}=\sqrt{2^{2p}}=\sqrt{(2^p)^2}=2^p$$
If $m$ is an odd positive integer we have:
$$\sqrt{2^m}=\sqrt{2^{2p+1}}=\sqrt{(2^p)^2\cdot 2}=2^p\sqrt 2$$
Therefore $\sqrt{2^m}$ will contain a radical when $m$ is an odd positive integer and will not contain a radical when $m$ is an even positive integer.