Long Expression Equals $a$

$\begin{aligned} \left(\frac x2\right)^{2x} & = x^x & \small \color{#3D99F6} \text{Raise both sides to the power of }\frac 1x \\ \left(\frac x2\right)^2 & = x \\ \frac {x^2}4 & = x & \small \color{#3D99F6} \text{Divide both sides by }x \\ \frac x4 & = 1 \\ \implies x & = 4 \end{aligned}$

Then, $\dfrac {\left(\frac x2\right)^\frac x2 (x^x) (2x)^{2x}}{(2x)^x} = \left(\dfrac x2\right)^\frac x2 (x^x) (2x)^x = \left(\dfrac 42\right)^\frac 42 (4^4) (8^4) = 2^{2+8+12} = \boxed{4194304}$

$\large \left( \dfrac{x}{2} \right)^{2x}=x^x \implies \left(\dfrac{x}{2}\right)^2=x ~~~~\color{#3D99F6}\small \text{(because x is a positive integer)}$ Therefore , $\large \dfrac{x^2}{4}=x \implies x=4~~~~\color{#3D99F6}\small \text{(since x is positive)}$

Hence $\large \dfrac{\left(\frac{4}{2}\right)^{\frac{4}{2}} \times 4^4 \times (2\times 4)^{2\times 4}}{(2\times 4)^4}=a$

Therefore $\large a=\dfrac{4 \times 256 \times 8^8}{8^4}=2^{22}=\boxed{4194304}$