Normally you can write $x*x=x^2$
But if you are operating within exponents, $a^{x*x} \neq a^{x^{2}}$ as the latter is equal to $a^{2x}$.
Is it a problem of notation ?
[Edited]
Thank you to having helped me to solve this problem. It is indeed a problem of notation, in the sense that parenthesis matters. From wikipedia: "Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up."
$$a^{x*x} = a^{x^{2}} = a^{(x^{2})} \neq (a^{x})^{2} = a^{x*2}$$
The expression $a^{x\cdot x}$ is equal to $a^{x^2}$ because $x\cdot x = x^2$.
However, $a^{x} a^{x} = a^{x+x} = a^{2x}$ because
$$\underset{x \text{ times} }{(a \cdots a )}\underset{x \text{ times} }{(a \cdots a)} = \underset{2x \text{ times} }{(a \cdots a)}. $$