Question

sqrt(72+36e^(12t)+36e^(-12t))=?

Answer 1

@phi

Answer 2

what do you want to do with this?
we could re-write it using cosh
\[ \cosh(x) = \frac{ e^x + e^{-x}}{2} \]
to get
\[ \sqrt{72+ 72 \cdot \frac{ e^{12t} + e^{-12t}}{2} } \\
\sqrt{72+ 72 \cosh(12t) }
\]

there are these identities
https://en.wikipedia.org/wiki/Hyperbolic_function#Useful_relations
in particular:
\[ \cosh(2x) = 2 \cosh^2(x) -1 \]
to write
\[ \cosh(12t) = 2 \cosh^2(6t) - 1 \]
factor out 36 from the square root:
\[ 6 \sqrt{2+ 2 \cosh(12t) } \\
6 \sqrt{2+ 2(2 \cosh^2(6t) - 1)} \\
6 \sqrt{4\cosh^2(6t)}\\
12 \cosh(6t)
\]
if the cosh is positive.