Dividing power towers by exponents

Question

Say we have $e^{e^{e^{e^e}}}$. Since exponents raised to exponents is the same as multiplying them, this is equivalent to $e^{4e}$:

$$e^{e^{e^{e^e}}}=e^{4e}$$

Factoring out an $e^e$:

$$e^{e^{e^e}}=e^4$$

The left side now collapses to $e^{3e}$, leading to the equality $3e=4$.

Clearly my mistake was in the part where I divided both sides by $e^e$, but why am I not able to do this?

Answer 1

It's because $e^{e^{e^{e^e}}}$ isn't $e^{4e}$ in the first place (and neither is $((((e^e)^e)^e)^e)$, which is $e^{e^4}$). The left expression is a power tower, where exponentiation happens right-to-left, not left-to-right as required for the power multiplication rule $(a^b)^c=a^{bc}$.

Answer 2

Note that

$$e^{e^{e^{e^e}}}\neq e^{4e}$$

as

$$3^{3^3} = 3^{27} \neq 3^9$$