Big product

Answer: 14

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

$$$\begin{array}{rl}{2}^{18}\times 5^{12}& =2^6\times 2^{12}\times 5^{12}\\ & =2^6\times (2^{12}\times 5^{12})\\ & =2^6\times {(2\times 5)}^{12}\\ & =2^6\times {10}^{12}\\ & =2^6\text{ with 12 zeros afterwards}\\ & =\underset{\text{2}}{\underbrace{64}}\underset{\text{12}}{\underbrace{00...0}}\end{array}$$$

So total $14$ zeros