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

When performing operations with numbers in scientific notation, the process differs for multiplication and division. For multiplication, you multiply the coefficients (the numbers in front of the exponential part) and add the exponents. For example,

$(A\times {10}^x)\times (B\times {10}^y)=(A\times B)\times {10}^{x+y}$

In division, you divide the coefficients and subtract the exponents, such as

$(A\times {10}^x)÷(B\times {10}^y)=(A÷B)\times {10}^{x-y}$

It's essential to remember that the final answer must have the least number of significant figures from the coefficients used in the calculation. Additionally, ensure that the coefficient in the scientific notation is between one and less than ten, adjusting the exponent accordingly if you move the decimal point. For instance,

$10.4822\times {10}^8\to 1.0\times {10}^9$

to maintain proper scientific notation with the correct number of significant figures.

Multiplication and Division Operations

So in this video we're going to take a look. What exactly do we do when we have sets of numbers and scientific notation and we're 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&B. And then we're going to add the exponents, which are our powers. Here our powers are X&Y.

So when we're doing a * 10 to the X * b * 10 to the Y, what we're doing here is we're multiplying A&B together and then we're going to add the exponents together, so X + y. This would be the answer to our expression. Now, when you divide the values in scientific notation, you're going to divide the coefficients, and you're going to subtract the exponents. So here we're going to have A being divided by B times 10 and now we have 10 to the X / 10 to the Y. So that becomes X -, y.

Now we're going to say we have to remember this after multiplying and Oregon dividing. Remember that the that for the coefficients we will have the least number of significant figures. So least number of sig fakes when you multiply or divide, its least number of sig figs when you add or subtract its 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 a different examples that we have here.

Here we have examples one and two. We're going to apply what we learned up above to answer them. We're 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 one 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're going to do here is we're going to multiply the coefficients here and then remember the exponents get added together O. When we multiply those 3 coefficients together of 2.134 * 1.66 * 3.07, we get 10.4822. And then remember the exponents are adding together. So that would be 5 + -3 + 6. So that would give me 8 at the end.

But remember when it comes to multiplying your coefficients, when you're doing multiplication or division, it has to be least number of significant figures. Now when you have a decimal place, remember we go from left to right, we start counting once you get to your first non 0 number. Your first non 0 number here is 2, so you start counting there and you count all the way until the end SO1234. Here's your first nonzero 1/2. Here's your first non 0123, so at least number of sick figs would be two.

Now, the way we look at this answer, it's not written correctly in terms of scientific notation. We remember your coefficient has to be a number between 1:00 but less than 10. So we're going to have to move this decimal over one. We just made our number smaller, it went from 12:50. If your coefficient gets smaller, that means your exponent's going to get larger, so it becomes 10 to the nine, and we need 2 SIGFIX. So this would be 1.0 * 10 to the nine. If you punch this into your calculator, you should get this answer.

Now remember, when you're 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're 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. So we have a combination of multiplication and division occurring in this question. So when it comes to the coefficients, here are our coefficients. We're multiplying those coefficients together here. When we do, that gives us 11.8454.

And then here the X bones, when they're multiplying, would be adding together, so that'd be 8 + -1, which gives me A7. But then you're dividing 10 to the seven by 10 to the 11. Remember, when you're dividing, it really means you're subtracting. So that give us -4. Our answer here is not written correctly. Your coefficient has to be a number between 1:00 and 10:00. So again, I'm going to move this decimal over one 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. And 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 three sick figs, this one here also has three sig figs, and then finally this one also has three sig figs.

So our answer at the end has to have three significant figures, so it'd be 1.18 * 10 to the -3 as my final answer.