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5. BONUS: Mathematical Operations and Functions

**Multiplication** and **division** of values in scientific notation.

Multiplication and Division

Whenever multiplying or dividing our final answer will have the least number of significant figures.

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So in this video, we're gonna take a look. What exactly do we do when we have a sense of numbers and scientific notation and we're multiplying or dividing them? Now we're gonna say When you multiply values and scientific notation, you multiply the coefficients. Remember, your coefficients are these values here in this case, it 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're doing a times 10 to the x times B times 10 to the Y. What we're doing here is we're multiplying a and B together, and then we're gonna add the exponents together. So x plus y, this would be the answer to our expression. Now, when you divide the values in scientific notation, you're gonna divide the coefficients and you're going to subtract the exponents. So here we're gonna have a being divided by B Times 10. And now we have 10 to the X, divided by 10 of the Y. That becomes X minus y. Now we're going to say we have to remember this after multiplying and dividing. Remember that that for the coefficients we will have the least number of significant figures, so least number of sig figs when you multiply or divide its least number of sick things when you add or subtract its least number of decimal places. Now that we've seen the basics in terms of multiplying 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 gonna apply what we learn up above toe. Answer that we're gonna 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.

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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 application. What we're gonna do here is we're gonna multiply the coefficients here and then remember, the exponents get added together. So we multiply those three coefficients together off 2.134 times 1.66 times 3.7 We get 10.4822 And remember, the exponents are adding together so that be five plus negative three plus six. So that would give me eight at the end. But remember, when it comes to multiply 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 zero number, your first non zero number here is to So you start counting there and you count all the way until the end. So 1234 Here's your first non 012 Here's your first non So at least number of sick things would be to. 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 one but less than 10. So we're gonna have to move this decimal over one. We just made our number smaller. It went from 10 to 1. If you're coefficient gets smaller, that means you're exponents gonna get larger. So becomes 10 to the nine and we need to six. Fix for this would be one point zero times 10 to the nine. If you punch this into your calculator, you should get this answer now. Remember, when you're doing a number of times 10 toe 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.

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so here it says, using the methods discussed above determined 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, your are 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 the eight plus negative one, which gives me a seven. But then you're dividing 10 to the seven by 10 to the 11th. Remember, when you're dividing, it really means you're subtracting, so that gives us negative four. Our answer here is not written correctly. Your coefficient has to be a number between one and 10. So again, I'm gonna 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 negative for now, it's gonna be negative. Three. And remember, we got our answer here, but it's not the best answer we're multiplying or dividing the coefficients. It's the least number of significant figures. So here, this one has three sig figs. This one here also has three sig figs. And then finally, this one also has 36 fix. So our answer at the N s have three significant figures. So it be 1.18 times 10 to the negative three as my final answer.

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