Is there a way to write this term different $e^{x+e^x}$?

Question

I am a little confused what this actually means:

$e^{x+e^x}$

It is obviously not the same if I for example $$e^{x}:= \lambda \\ e^{x+e^x} \neq \lambda^\lambda $$

Answer 1

It is the same as $$e^x\cdot e^{e^x}$$

Answer 2

You're going wrong slightly in the last step:

$$e^x = \lambda$$

$$\implies e^{x+e^x} = e^x e^{e^x} = \lambda \times e^{\lambda}$$

Your expression $$\lambda^{\lambda} = (e^{x})^{e^x}$$

See the difference?