From Förberedande kurs i matematik 1

The partial expression \displaystyle 2^{2^{3}} should be interpreted as \displaystyle 2 raised to the \displaystyle 2^{3},

and because \displaystyle 2^{3}=2\centerdot 2\centerdot 2=8, thus \displaystyle 2^{2^{3}}=2^{8}

In order to calculate the next part of the expression, \displaystyle \left( -2 \right)^{-4},

it can be useful to do it a step at a time:

\displaystyle \begin{align} & \left( -2 \right)^{-4}=\frac{1}{\left( -2 \right)^{4}}=\frac{1}{\left( \left( -1 \right)\centerdot 2 \right)^{4}}=\frac{1}{\left( -1 \right)^{4}\centerdot 2^{4}} \\ & \\ & =\frac{1}{1^{4}\centerdot 2^{4}}=\frac{1}{2^{4}}=2^{-4} \\ \end{align}

Thus,

\displaystyle 2^{2^{3}}\centerdot \left( -2 \right)^{-4}=2^{8}\centerdot 2^{-4}=2^{8-4}=2^{4}=16