Big product

Answer: 14

$2\times5$=$10$ and multiplying by $10$ adds a digit (a zero at the end)

$\begin{split}2^{18}\times5^{12}&=2^6\times2^{12}\times5^{12}\\

&=2^6\times\left(2^{12}\times5^{12}\right)\\

&=2^6\times\left(2\times5\right)^{12}\\

&=2^6\times10^{12}\\

&=2^6\text{ with 12 zeros afterwards}\\

&=\underbrace { 64}_{\text{2}}\underbrace{00...0}_{\text{12}}\end{split}$

So total $14$ zeros