$2^{x^{\cos(x)}}\sqrt{\cos(x)}$ can you rearrange mathematically to ${\cos(x)}\sqrt2^{x^{\cos(x)}}$

Question

$2^{x^{\cos(x)}}\sqrt{\cos(x)}$ can you rearrange mathematically to ${\cos(x)}\sqrt2^{x^{\cos(x)}}$ if $x > 0$ and $\cos(x) > 0$

Answer 1

They are not the same. To show so we note that that there is a problem at $x=2\pi$, because we have $$2^{(2\pi)^{\cos(2\pi)}}\sqrt{\cos(2\pi)}=2^{2\pi}$$ while $$\cos(2\pi)\sqrt{2}^{(2\pi)^{\cos(2\pi)}}=\sqrt{2}^{(2\pi)}=2^{\pi}$$

For more evidence that these functions are not the same, here is part of a graph of the two functions.