Big product
How many digits are there in this product?
How many digits are in the product $2^{18}$ x $5^{12}?$
This problem is taken from the World Mathematics Championships
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