Why does $\sqrt{8^2+(8\sqrt 3)^2}=16$
\begin{equation} \begin{split} \text{n}&:=\sqrt{8^2+(8\sqrt{3})^2}\\ \\ &=\sqrt{64+(8\sqrt{3})^2}\\ \\ &=\sqrt{64+8^2\cdot\left(\sqrt{3}\right)^2}\\ \\ &=\sqrt{64+64\cdot 3}\\ \\ &=\sqrt{64+192}\\ \\ &=\sqrt{256}\\ \\ &=\sqrt{16^2}\\ \\ &=16 \end{split}\tag1 \end{equation}
Alternative to Jan Eerland his answer:
\begin{align} \sqrt{8^2 + (8\sqrt{3})^2} = \sqrt{8^2 + 8^2 (\sqrt{3})^2} = \sqrt{8^2 + 8^2 \cdot 3} = \sqrt{8^2(1 + 3)} = \sqrt{8^2 \cdot 2^2} = 8 \cdot 2 = 16. \end{align}