Here's an approach. It assumes familiarity with logarithms.
Also, it's handy to memorise log2 = 0.3, log3=0.48, log5=0.7
All logs are on a base of 10.
If x= 12^12 x 15^18
logx =log(12^12 x 15^18)
= log(3^12 x 4^12 x 3^18 x 5^18)
= log(2^24 x 3^30 x 5^18)
= log(2^24) + log(3^30) + log(5^18)
= 24log2 + 30log3 + 18log5
= 24x0.3 + 30x0.48 + 18x0.7
x = 10^34.2
And that means that x has 35 digits
Thanks, I never thought of applying logarithms since it is not tested on GMAT. But in any case this is one way to solve the problem and it is nice. I wonder if there is any other way to atleast approximate the number of digits (without using the logarithms)