why are $e^{2x}$ and $e^{x^2}$ inequal?

Question

enter image description here

From the index rules I learned from school, $a^{x^2}=a^{2x}$ Does it work the same for the natural constant? Why is it?

Answer 1

There is a difference between ${(a^b)}^c$ and $a^{(b^c)}$.

The first simplifies to ${(a^b)}^c=a^{bc}$ but the second does not.

If no brackets are used, it is assumed that it means $a^{b^c}=a^{(b^c)}$, the one that does not simplify.

Answer 2

$ a^{x^2}=a^{2x }$ is not true for all $x$ !

We have

$$a^{x^2}=a^{2x } \iff a^{x^2-2x}=1 \iff x^2-2x=0 ,$$

Hence $a^{x^2}=a^{2x } \iff$ $x=0$ or $x=2.$

Answer 3

There is no reason to assume that exponentiation is associative; and in fact, it is not. (Counter example: $2^{2^3} = 2^8$ but $ (2^2)^3 = 2^6$) Thus in general, $a^{(x^2)}$ (which is what we usually mean when we write $a^{x^2}$ ) is not necessarily equal to $(a^x)^2 = a^{2x}$.