Learn the toughest concepts covered in your Analytical Chemistry class with step-by-step video tutorials and practice problems.

Experimental Error

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

When multiplying or dividing your final answer will have the least number of significant figures.

1

Multiplication and Division Operations

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So when you multiply values in scientific notation, remember that you multiply the coefficients and you're gonna add the exponents. So here we have eight times 10 to the X times b times 10 to the Y. So you multiply the coefficients together and then we add the exponents together. So it'd be X plus Y. Now when we're dividing we're gonna divide the coefficients and you're going to subtract the exponents. So here this would become a divided by B Times 10 X -Y. Now, after multiplying and or dividing, remember that the coefficients will have the least significant figures doing that will give us our our most basic answer, our most correct answer at the end of our calculations. So applying what we've just learned, try to solve example one come back and see how I approached that same problem. Remember use the methods that are described up above to try to answer this question correctly.

2

Multiplication and Division

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So here we're gonna say using the methods discussed above, we're gonna attempt to answer this question here. So remember when we're multiplying or dividing our coefficients need to have the least number of sig figs. So our coefficients are here. So when we multiply those three coefficients together we get initially 10.4, times 10 with our exponents here, we're gonna have to add them together. Five plus minus three plus six here. So five plus minus three plus six here gives us eight. So it's gonna be times 10 to the eight. Remember when it comes to your coefficient? It has to be a value that is between that is between one and 10, it has to be a value that's equal to or greater than one but less than 10. So I'm gonna move this this decimal point over one to make it one point something. by decreasing it by one, I have to increase my exponent by one. So now this becomes 1.4822 times 10 to the nine. Now our coefficients, this one here has four sig figs. This one here has two sig figs. And this one here has three sig figs. We want the least number of significant figures for coefficient. So I have to So it's gonna come 1.0 times 10 to the nine as our final answer here. So just remember to follow the steps and techniques that we've used thus far in order to get the correct answer. Not that you've seen that one attempt to do example to once you've done that come back and see how I approach that same exact question.

3

Multiplication and Division

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in this one. We have mixed operations but the methods are still the same. We're gonna first figure out. Well my answer will be when I multiply these two 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 to the eight times 10 to negative one. We're really adding those together. That's 10 to the seventh, divided by 10 to the 11th, Which is really 7 -11 which gives us -4. So that's gonna be times 10 to the -4. Remember we need to write our scientific notation where the coefficient is between one and 10. So I'm gonna move this decimal over one because it decreased by one. That means my exponent has to increase by one. So it becomes 1.18454 times 10 to the minus three. In terms of our coefficients here, this one has three significant figures. This one here has three significant figures and this one here also has three significant figures. So my answer at the end would be 1.18 times 10 to the -3. That's 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

4

Multiplication and Division Calculations

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So here we have to find the correct number of significant figures for the different examples provided. So these are mixed operations with a combination of addition, subtraction, division or multiplication that we have to employ. So 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 that's being involved in addition and subtraction. So for the top the larger value is the 10 to the negative three which means I need to manipulate this one So that it is also 10 to the -3. Now we need to increase this by two so it becomes 10 to negative three. So I need to make this coefficient smaller by two. So we move the decimal over by two. So when we do that We're gonna have .0912 times 10 to the -3 Plus 6.33 times 10 to the -3, Divided by what's on the bottom. So here 10 to the seventh is the larger value. So we're gonna have to manipulate this here so that it is also 10 to the seventh. So I'm gonna move this over by one. So when I move it over by one it becomes 10 to the seventh. So on the bottom we're gonna have 1.15 times 10 to the seventh -17, 2 times 10 to the 7th. Alright so now we can add those values on top to track those values values on the bottom. Remember in this case it needs to be the least number of decimal places. So here we have four decimal places. Here we have two decimal places. So my number at the end has to have two decimal places. So that would give us 6.42 times 10 to the -3 if we do it correctly divided by. So now we're gonna subtract These two coefficients. Again it's the least number of decimal places. This one has two whereas this one here has three so if we do that correctly we'll have two decimal places at the end so that's .78 Times 10 to the 7th at this point we have division that's being undertaken. So here it has to be least. Sig figs here. This coefficient has three sig figs. Whereas this one here only has two sig figs. So our number at the end has to have two sig figs. So that's gonna give me 8.2 Times 10. Remember when we're dividing we're actually subtracting these exponents? So it's gonna be negative 3 -7 which gives us negative 10. So the answer you should get here is 8.2 times 10 to the negative 10. So 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 this is the method you have to do. You have to make sure if you're adding or subtracting the exponents match up so that you could add the coefficients or subtract the coefficients, then remember it's at least number of decimal places. Once you get to multiplication or division, it would be least number of sig figs for a 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 one attempt this on your own. Don't worry. Just come back and see how I approach the same question to get the final answer.

5

Problem

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}

6

Problem

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

A

9.994×10^{21}

B

9.990×10^{21}

C

1.002×10^{22}

D

1.004×10^{22}

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