How many digits long is the number $2^{10}\times 5^7$ once it has been multiplied out?

Pair up the 2s and 5s to make 10s: $2^{10}\times 5^7=2^3\times (2^7\times 5^7)=8\times 10^7=80000000$. That is an 8 followed by seven zeros, so 8 digits in all. Counting only the seven zeros gives 7, and adding the indices 10 + 7 gives 17 — neither counts the leading 8 correctly.