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

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents. For division, divide the coefficients and subtract the exponents. Significant figures are crucial: for multiplication and division, use the least number of significant figures; for addition and subtraction, use the least number of decimal places. Ensure coefficients are between 1 and 10 for proper scientific notation. For example, multiplying 2.13 × 10⁵ by 1.66 × 10³ results in 3.54 × 10⁸ with appropriate significant figures.

Multiplication and Division Operations

So in this video, we are going to take a look at what exactly do we do when we have sets of numbers in scientific notation and we are multiplying or dividing them. Now, we're going to say when you multiply values in scientific notation you multiply the coefficients. Remember, your coefficients are these values here, in this case, it would be variables a and b. And then we're going to add the exponents which are our powers. Here our powers are x and y. So when we are doing a×10x×b×10y, what we are doing here is multiplying a and b together and then we are going to add the exponents together. So this would be the answer to our expression. Now, when you divide the values in scientific notation you are going to divide the coefficients and you are going to subtract the exponents. So here we are going to have a÷b×10x−y as the result of division.

After multiplying and dividing, remember that for the coefficients we will have the least number of significant figures. When you multiply or divide, it's the least number of significant figures. When you add or subtract, it's the least number of decimal places.

Now that we've seen the basics in terms of multiplying and dividing numbers in scientific notation, let's take a look at the different examples that we have here. Here we have examples 1 and 2. We are going to apply what we learned above to answer them. We are going to approach them in the same method that we talked about in the previous two examples. Now, come back, take a look at example 1 and how we approach it. If you want to do it ahead of time on your own, you can do that as well.

Multiplication & Division Calculation 1

Using the method discussed above, determine the answer when the following values are multiplied. So here we have these three values in scientific notation undergoing multiplication. What we are going to do here is we are going to multiply the coefficients and then remember the exponents get added together. So when we multiply those three coefficients together of 2.13, 1.66, and 3.07, we get 10.4822. And remember the exponents are adding together so that would be 5 + (-3) + 6, which gives a total of 8. But remember, when it comes to multiplying your coefficients, when you are doing multiplication or division it has to be the least number of significant figures. When you have a decimal place, remember we go from left to right. We start counting once you get to your first non-zero number. Your first non-zero number here is 2, so you start counting there, and you count all the way until the end. So 1, 2, 3, 4. Here is your first non-zero, 1, 2. Here's your first non-zero, 1, 2, 3. So the least number of significant figures would be 2.

Now the way we look at this answer, it's not written correctly in terms of scientific notation. Remember, your coefficient has to be a number between 1 but less than 10. So, we are going to have to move this decimal over 1. We just made our number smaller; it went from 101. If your coefficient gets smaller that means your exponent is going to get larger. So it becomes 10 to the 9. And we need 2 significant figures so this would be 1.0×10⁹. If you punch this into your calculator, you should get this answer. Now remember when you are doing a number times 10 to any power, you should put it in parentheses in your calculator. So you should put parentheses around each of these numbers and then multiply them. If you don't do that, your calculator most likely will give you the incorrect answer. But based on what we've seen up above, this would be the correct value when we are multiplying these three numbers in scientific notation. Attempt to do the following example just like we did this one. Come back and see if your answer matches up with mine.

Multiplication & Division Calculation 2

So here it says using the methods discussed above, determine the answer for the following Mixed Operations question. We have a combination of multiplication and division occurring in this question. When it comes to the coefficients, here are our coefficients, we are multiplying those coefficients together here. When we do that, it gives us 11.8454. And then, the exponents when they're multiplying would be adding together. So that would be 8 + (-1) which gives me 7. But then you're dividing 10 to the 7 by 10¹¹. Remember when you are dividing it really means you are subtracting, so that would give us negative 4. Our answer here is not written correctly. Your coefficient has to be a number between 1 and 10. So again, I'm going to move this decimal over 1 to give us 1.18454. We just made that coefficient smaller which means our power is going to become larger. So instead of being -4 now it's going to be -3. Remember we got our answer here but it's not the best answer. When we're multiplying or dividing the coefficients, it's the least number of significant figures. So here, this one has 3 significant figures, this one here also has 3 significant figures, and then finally, this one also has 3 significant figures. So our answer at the end has to have 3 significant figures,1.18×10−3 is my final answer.