How does $\frac{1}{2} \sqrt{4 + 4e^4} = \sqrt{1 + e^4}$ My understanding would lead me to believe that:
$$\frac{1}{2} \sqrt{4 + 4e^4} = \frac{1}{2}(2 + 2e^4) = 1 + e^4$$
But it actually equals: $\sqrt{1 + e^4}$
Can you explain why?

Also you can use squaring both sides:
$$
\sqrt{4+4e^4}=2\sqrt{1+e^4}\\
4+4e^4=4(1+e^4)
$$
since  $\sqrt{x^2}= \pm x$ and the expressions you start with is one of these two solutions (positive)
