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 are in your number with the least amount of significant figures. For example if you were multiplying 40.6 by 19.14 your number with the least amount of significant figures would be 40.6 with 3 significant figures verses the 4 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 was 3, we would round the product to three significant figures. This makes your final answer 777
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