Explain this factorization: $\sqrt{1+e^x} - \sqrt{e^x}=\sqrt{e^x}(\sqrt{e^{-x}+1}-1)$

Question

I found this factorization in one of my math books: $\sqrt{1+e^x} - \sqrt{e^x}=\sqrt{e^x}(\sqrt{e^{-x}+1}-1)$

I don't understand how that is possible. Can someone explain what rules are used for this factorization? Thanks.

Answer 1

Use:

$$\sqrt{1 + e^x} = \sqrt{\underbrace{(e^{-x} e^x}_{1}) (1 + e^x)} = \sqrt{e^x}\sqrt{e^{-x}+1}$$