Here's how I looked at it,

(4^9)(5^17)
= ((2^2)^9)(5^17)
= (2^18)(5^17)
= 2*(2^17)(5^17)
= 2* (2*5)^17
= 2 * 10^17

From observation we know that whenever 10 is raised to a power n, the number of digits is n+1
10^1 #digits 1+1
10^2 #digits 2+1
10^3 #digits 3+1
etc.

Therefore, 10^17 will have 17+1= 18 digits
The multiplier 2 just doubles the number and does not add to the number of digits.