Multiplication and Division Operations - Online Tutor, Practice Problems & Exam Prep

When multiplying values in scientific notation, multiply the coefficients and add the exponents: \( a \times 10^x \times b \times 10^y = (a \times b) \times 10^{(x+y)} \). For division, divide the coefficients and subtract the exponents: \( \frac{a}{b} \times 10^{(x-y)} \). Always ensure the final answer reflects the least number of significant figures from the coefficients. This method is crucial for accurate calculations in chemistry, particularly in stoichiometry and thermodynamics.

Multiplication and Division Operations

When you multiply values in scientific notation, remember that you multiply the coefficients and you're going to add the exponents. So here we have a⋅10x⋅b⋅10y. So you multiply the coefficients together and then we add the exponents together. So it'd be x+y. Now, when we're dividing, we're going to divide the coefficients and you're going to subtract the exponents.

So here, this would become ab⋅10x-y. After multiplying and or dividing, remember that the coefficients will have the least significant figures. Doing that will give us our most basic answer, our most correct answer at the end of our calculations. Applying what we've just learned, try to solve example 1. Come back and see how I approach that same problem.

Remember, use the methods that are described up above to try to answer this question correctly.

Multiplication and Division

So here, we're going to say, using the methods discussed above, we're going to attempt to answer this question here. So remember, when we're multiplying or dividing, our coefficients need to have the least number of significant figures. So our coefficients are here. So when we multiply those three coefficients together, we get initially 10.4822×101. With our exponents, we're going to have to add them together.

5+3+6 here. So 5+3+6 gives us 14. This is going to be times 1014. Remember, when it comes to your coefficient, it has to be a value that is between 1 and 10. It has to be a value that's equal to or greater than 1 but less than 10.

I'm going to move this decimal point over 1 to make it 1 point something. By decreasing it by 1, I have to increase my exponent by 1. So now this becomes 1.04822×109. Now our coefficients, this one here has 4 significant figures. This one here has 2 significant figures, and this one here has 3 significant figures.

We want the least number of significant figures for our coefficient, so we'd have 2. So it's going to come 1.0×109 as our final answer here. Just remember to follow the steps and techniques that we've used thus far in order to get the correct answer. Now that you've seen that one, attempt to do example 2. Once you've done that, come back and see how I approach that same exact question.

Multiplication and Division

In this one, we have mixed operations but the methods are still the same. We're going to first figure out what my answer will be when I multiply these 2 coefficients then divide by this one here. So when we do those methods, we get 11.8454. And then remember, when you're multiplying, you add the exponents together and when you're dividing, you subtract them. So multiplying times \(10^8\) times \(10^{-1}\), we're adding those together.

That's \(10^7\) divided by \(10^{11}\) which is \(7 - 11\) which gives us \(-4\). That's going to be times \(10^{-4}\). Remember, we need to write our scientific notation where the coefficient is between 1-10. I'm going to move this decimal over 1. Because it decreased by 1, that means my exponent has to increase by 1.

So it becomes \(1.18454 \times 10^{-3}\). In terms of our coefficients here, this one has 3 significant figures. This one here has 3 significant figures, and this one here also has 3 significant figures. So my answer at the end would be \(1.18 \times 10^{-3}\) as my final answer. So just remember when we're multiplying versus when we're dividing, what are the proper operations that we need to do in order to get our final answer?

Multiplication and Division Calculations

So here we have to find the correct number of significant figures for the different examples provided. These are mixed operations with a combination of addition, subtraction, division, or multiplication that we have to employ. We have to remember all the concepts that we learned prior to be able to answer these questions. If we take a look at the first one, we have addition happening on top, whereas we have subtraction happening on the bottom. In both cases, remember, we need to manipulate them so that we have the same exponent involved in addition and subtraction.

For the top, the larger value is 10 to the negative 3, which means I need to manipulate this one so that it is also 10 to the negative 3. Now we need to increase this by 2, so it becomes 10 to the negative 3. So I need to make this coefficient smaller by 2. So we move the decimal over by 2. So when we do that, we're going to have 0.0912 times 10^-3 plus 6.33 times 10^-3 divided by what's on the bottom.

So here, 10 to the 7th is the larger value. So we're going to have to manipulate this here so that it is also 10 to the 7th. So I'm going to move this over by 1. So when I move it over by 1, it becomes 10 to the 7th. So on the bottom, we're going to have 1.15 times 10⁷ minus 0.372 times 10⁷.

Alright. So now we can add those values on top, subtract those values on the bottom. Remember, in this case, it needs to be the least number of decimal places. Here, we have 4 decimal places. Here, we have 2 decimal places.

My number at the end has to have 2 decimal places. That would give us 6.42 times 10^-3 if we do it correctly. Divided by so now we're going to subtract these 2 coefficients. Again, it's the least number of decimal places. This one has 2 whereas this one here has 3.

If we do that correctly, we'll have 2 decimal places at the end. So that's 0.78 times 10⁷. At this point, we have division that's being undertaken. So here it has to be the least significant figures. Here this coefficient has 3 significant figures whereas this one here only has 2 significant figures.

So our number at the end has to have 2 significant figures. That's going to give me 8.2 times 10. Remember when we're dividing, we're actually subtracting these exponents. So it's going to be negative three minus seven, which gives us negative ten. So the answer you should get here is 8.2 times ten to the negative ten.

Remember, this is important because we could simply put these numbers in a calculator to get the correct answer. But if on an exam you're asked to show step by step the process that you would use in order to isolate your final answer, this is the method you'd have to use. You'd have to make sure if you're adding or subtracting that the exponents match up so that you could add or subtract the coefficients. Then remember, it's the least number of decimal places. Once you get to multiplication or division, it would be the least number of significant figures for our final answer.

Now that you've seen this one, do a step-by-step process to find the correct answer with the right number of significant figures for practice question 1. Attempt this on your own. Don't worry. Just come back and see how I approach the same question to get the final answer.

Perform the following calculation for the image below to the right number of sig figs:

[(11.422-0.800) + (8.0 + 1.115)/0.0720] 1.33 x 10^-5

• A. 3.64×10^-3
• B. 3.639×10^-3
• C. 3.6×10^-3
• D. 3.60×10^-3

Compute the following and determine the correct number of significant figures in the answer for the image below: 

• A. 9.994×10²¹
• B. 9.990×10²¹
• C. 1.002×10²²
• D. 1.004×10²²