polar coordinates
So I know in polar coordinates the function such as $\cos (2 \theta)$ can be easily mapped and furthermore can be mapped on graphing softwares such as Desmos or Wolfram Alpha.
But how come these softwares will not let me graph polar functions like $\cos^{0.18}(3\theta)$?
I'd say that it's convenient to include it in the set of rose curves—it can be useful to think about $r=a\sin n\theta$ for noninteger values of $n$ (even starting at $0$), rather than just integers greater than $1$:
You can simply use the standard transformation from Cartesian coordinates to polar coordinate:
Given $r>0\;\text{and}\;\theta\in [0,2\pi)$
$$x=r\cos(\theta),\tag{$x$}$$ $$y=r\sin(\theta)\tag{$y$}$$
You are given that $$r=2\cos (3\theta)\tag{$r$}$$ So, replacing $r$ in each of the equations $(x)$ and $(y)$ with its equivalent $r = [2\cos(3\theta)]$, gives you:
$$x=[2\cos (3\theta)] \cos (\theta)\;\;\text{and}\;\; y=[2\cos (3\theta)] \sin (\theta).$$
Here's a nice image to help make sense of the "standard transformation" from Cartesian to polar coordinates: The Cartesian point $(x, y)$ is the furthest point from the origin along the blue line (length $r$), so given $\theta$ and the radius $r$, $(x, y) = (r\cos \theta, r \sin\theta)$.
It also helps to note the right triangle: $$\cos\theta = \dfrac{x}{r} \iff x = r \cos\theta\;\text{ and}\;\sin\theta = \dfrac yr \iff y = r\sin\theta.$$ This image is a good reminder as to how to transform $x, y$ into polar coordinates.
See also Polar coordinate system.
Best Answer
Patrick was right in his comment; Desmos is perfectly happy to graph $r_1=(\cos(3\theta))^{0.18}$, but it won't graph $r_2=\cos^{0.18}(3\theta)$, even though a human being would likely understand what you mean by the second form. Desmos will only graph functions involving terms of the form $\cos^{\hspace{0.4mm}n}(x)$ and/or $\sin^n(x)$ if $n=2$ or $n=-1$ (and in the latter case, Desmos assumes you want the inverse sine or cosine functions, which may not necessarily be true).
For reference:
Wolfram Alpha has similar preferences; it tends to prefer functions of the form that $r_1$ is in, no matter to what exponent you want to raise a sine or cosine.