How to understand $2^{e^{x}}$?

Question

$2^{e^{x}}$ is an exponent over a exponent. It is confusing. How to understand it? Can I simplify it?

Answer 1

No, you can't simplify it. It increases very rapidly with $x$. We read it as $2^{(e^x)}$ because $(2^e)^x=2^{ex}$

Answer 2

The way exponent towers work is you go from the top down. So: $$4^{3^2}=4^9=262,\!144$$ $$2^{e^4}\approx2^{54.59815}\approx27,\!269,\!731,\!868,\!896,\!801.97168$$ $$2^{3^4}=2^{81}=2,\!417,\!851,\!639,\!229,\!258,\!349,\!412,\!352$$ $$10^{10^2}=10^{100}=10,\!000,\!000,\!000,\!000,\!000,\!\\ 000,\!000,\!000,\!000,\!000,\!000,\!000,\!000,\!000,\!000,\!\\ 000,\!000,\!000,\!000,\!000,\!000,\!000,\!000,\!000,\!000,\!\\ 000,\!000,\!000,\!000,\!000,\!000,\!000,\!000$$

It's interesting how $2^{e^4}$ is around 30 million billion, but $2^{3^4}$ is around 2 million billion billion (that is, around 70 million times larger).