Indices Question & Equation

Question

Following is the equation

$$x^{x\sqrt x}=x\sqrt{x}$$

We need to find $x$. Please help.

Answer 1

HINT

Take logs to get $$ x\sqrt{x} \ln x = \ln x + \frac12 \ln x \\ \ln x \left(x \sqrt{x} - \frac32\right) = 0 $$

Can you finish?

Answer 2

What if $x=0$?

For finite non-zero $x,$

$$x^{x^{3/2}-3/2}=1$$

From Find all real numbers $x$ for which $\frac{8^x+27^x}{12^x+18^x}=\frac76$, check if $u^m=1$