Why, when solving this problem in an equation we have to associate or disassociate the e^x as a unit because when we solve and separate out the x^e or even x^(e^x) as a unit the answer is different and wrong. Why does this happen? If this is understood as ((e^x^)^(e^x)) the answer is correct because this does not equal e^((x^e)^x). Is there another layer of order of operations involved here for the term e^x?
Background: I was told this in high school while I was learning calculus to just accept this but it never made sense? I was told that e^x has to stay together and it is the way it is because we do it that way.
It's $$\mathrm{e}\text{^}x\text{^}\mathrm{e}\text{^}x = \mathrm{e}\text{^}(x\text{^}(\mathrm{e}\text{^}x)) = \mathrm{e}^{x^{\mathrm{e}^x}} \text{.}$$ Exponentiation is right-associative: in the form $a\text{^}b\text{^c}$ the right-most exponent "binds more tightly", so must be evaluated first. This applies recursively, so in a chain of exponents, we parenthesize as written above.
(Aside: $$(\mathrm{e}\text{^}x)\text{^}(\mathrm{e}\text{^}x) = \left( \mathrm{e}^x \right)^{\mathrm{e}^x}$$ is neither left nor right associative. It is not the rule used for evaluating exponentials.)
Notice that pretending that exponentiation were left-associative would produce different results. \begin{align*} (2\text{^}3)\text{^}4 &= (8)\text{^}4 = 4096 \text{ and} \\ 2\text{^}(3\text{^}4) &= 2\text{^}81 = 2\,417\,851\,639\,229\,258\,349\,412\,352 \text{.} \end{align*} Another way to write this is $$ (2^3)^4 = 2^{3 \cdot 4} = 2^{12} = 4096.$$ That is, if we pretend that exponentiation is left-associative, we must end up multiplying all the exponents together, instead of exponentiating them. And then there's no reason to use the stacked notation "$\mathrm{e}^{x^{\mathrm{e}^x}}$" if we just wanted $\mathrm{e}^{x \cdot \mathrm{e} \cdot x}$ So in order that a repeated exponential not be "just ordinary multiplication of the exponents", exponentiation has to be right associative.