When multiplying measurements in chemistry we quite often can't get exact measurements. Either that or the measurements we do get have so many figures we would not be able to effectively write them all out. This is when we round using significant figures.
How to Use Significant Figures in Multiplication and Division
Determine if your numbers are measurements. If they are, they are probably going to be inexact. This is when you would need to round using significant figures.
Determine how many significant figures, within the numbers you are multiplying or dividing, contains the least amount of significant figures. For example if you were multiplying 40.6 by 19.14, the number with the least amount of significant figures would be 40.6 with 3 significant figures verses the 4 significants in 19.14.
Round your solution to the amount of significant figures you discovered in step 2. If we were to multiply 40.6 by 19.14 you would get a product of 777.084. Since the least amount of significant figures we discovered was 3, we would round the product to three significant figures. This makes your final answer 777.