Answer     $64$

We have $$x^x = 2^{3x}\cdot2^{192}=8^x\cdot 2^{192}\implies \left(\frac{x}{8}\right)^x = 2^{192}\implies \left(\frac{x}{8}\right)^{\frac{x}{8}}=2^{\frac{192}{8}} $$

Therefore, we have $$\left(\frac{x}{8}\right)^{\frac{x}{8}}=2^{24}\implies \left(\frac{x}{8}\right)^{\frac{x}{8}} = 8^8 \implies \frac{x}{8}=8\implies x=\boxed{64}$$