 Greetings and welcome to Math Help for Science courses. In this video I am going to go through and discuss scientific notation calculations. So how do we go about doing calculations using scientific notation? So what we want to do first is to review a little bit about what we mean by scientific notation. So let's quickly take a look at that. And what we mean with scientific notation is that this is a way to be able to conveniently express both very large and very small numbers. In science we often have numbers that are much larger than we are typically used to using or that are much smaller. And because of all of those zeros involved in those it makes it much more convenient to express them in scientific notation which is based on powers of ten. So when we set up a number in scientific notation we split it into two parts and these are going to be multiplied together. So we take the digits term and the exponential term. And let's look at a quick example of this. If we take the number ninety three million not too bad to write out but it still does have all of those zeros. We write that as nine point three times ten to the seventh power. So we've split this up and it gets much more convenient to write these numbers and to be able to do calculations with them as we will see today. In this case we have the digits term here nine point three and the exponential term is ten to the seventh power. So let's look at how we go about doing some calculations with these numbers. And we start off here. What we really want to do is work with each term individually. So we don't want to work with them together. We're not going to work with them all at once. We're going to work with them individually. So we're going to work with each part of the number in scientific notation. And what we want to do is to first look at the digits term and get those combined. Then we're going to combine the exponential terms together. And finally we want to bring it back into what we call standard notation meaning that there is just one number to the left of the decimal point. And I should say one non-zero number to the left of the decimal point when the final number is written. Now these can be done very easily using a scientific calculator and you can do it all at once putting the numbers in there. And I will show you briefly how that can be done towards the end of this lecture. But let's look at doing a couple of them by hand first. There's a little bit more to it this and it depends on whether you are adding the numbers or subtracting the numbers or whether you are multiplying and dividing. There are a couple of different steps. So let's look at examples of each of those two. But first if we want to look at an example of addition or subtraction let's look at the steps that we need to take. So if we are going to add or subtract numbers first of all we have to make the exponents the same. If the exponents are not the same you cannot add or subtract them. So we have to let's look at two examples here. Let's look at our first example and that is 7.5 times 10 to the third plus 4.8 times 10 to the third. Now in this case the exponential portion is already the same so there is nothing else to do. We can just leave those alone as they are and add them together. So we would have 4.8 plus 7.5 gives you 12.3. So we just add those together and then our answer would be 12.3 times 10 to the third power. However this is not in standard form. So what we want to do is to convert this into standard form and that means we want only one non-zero digit to the left of the decimal point. So we have to move this digit one place and that means that we then have to adjust the exponent a little bit and we have to change the 3 into a 4. So our final answer would then be 1.23 times 10 to the fourth power. So if we add 7.5 times 10 to the third plus 4.8 times 10 to the third we would get 1.23 times 10 to the fourth power. Now let's look at an example where we do have to change that and we'll be looking at a subtraction example as well. So here we have 4.7 times 10 to the negative fifth power minus 1.8 times 10 to the negative sixth power. Now we have to convert these to be the same exponent. So that means we have to change one of these numbers to make the exponent correct. So we can do this with either one of them. It does not matter. You can change either one whatever you find more convenient. Let's try changing this one and let's move our decimal point one place to the right. That will make our exponent more negative so our exponent will go from a negative 5 to a negative 6 which is what we want to match this one. So then we can write this number instead as 47 times 10 to the negative sixth power minus 1.8 times 10 to the negative sixth power. Now the exponent is going to be the same so all we have to do is do the subtraction. 47 minus 1.8 would give us 45.2. So our answer would then be 45.2 times 10 to the negative sixth or converting it into standard notation. So let's move the decimal point back one place to the left that increases our exponent from negative 6 to negative 5 and therefore gives us our final answer which would be 4.52 times 10 to the negative fifth. So those are just a couple of examples of how you can go about adding the numbers. So let's write our answers over here for the bottom one we just did that that was 4.52 times 10 to the negative fifth. For our first problem when we added the two numbers together we got 1.23 times 10 to the fourth power. Now that's how we do addition and subtraction. Let's look at some examples of how we would do multiplication and division. Now when we're going to multiply or divide these numbers it's a little bit different than adding or subtracting. What we want to do is to first of all multiply the digits together and then we have to either add or subtract the exponents depending on whether we are multiplying or dividing. If we are multiplying the numbers together we would add the exponents. If we are dividing them then we would subtract the exponents and of course as always we reduce it to standard form. So let's look at our first example. We of course first of all we multiply the digits so 5.38 times 2.58 is equal to 13.9. And that's what we do those just using again just the digits portion all we did was multiply them together. Now in this case we are multiplying so when we are multiplying 10 to the seventh power times 10 to the negative fourth power we are going to add the exponents so that becomes 10 to the power of 7 plus a negative 4. Now if you are adding a negative number that is the same as subtracting so that would be 7 plus a negative 4 is the same as 7 minus 4 or this is 10 to the third power. So that would give us our answer as 13.9 times 10 to the third power. However it is not in standard form so let's adjust that again and we want to move our decimal point one place to the left change our exponent increase it by one because we moved to the left and that would give us our final answer and that would give us our final answer of 1.39 times 10 to the fourth power. So this problem would equal 1.39 times 10 to the fourth. So let's clear this so we can look and do our last example here and now let's look at our final example and what we want to do now is divide two numbers we have 2.86 times 10 to the negative third and we want to divide that by 8.58 times 10 to the sixth so first we divide the digits so 2.86 divided by 8.58 is equal to 0.333. Now we have to do the exponents if we are dividing then we want to subtract the exponents so we have 10 to the minus third and we're dividing that by 10 to the sixth so that is going to become 10 to the negative 3 minus a positive 6. So negative 3 minus 6 becomes negative 9 so this becomes 10 to the negative 9th power and now we will have our final answer of 0.333 times 10 to the negative 9th. However as with the previous one we do have to convert that back into standard notation so our decimal point in this case we do not have a non-zero number to the left so we have to move our decimal point one place to the right that makes our exponent change by 1 so it's going to go from negative 9 to negative 10 and our final answer once we've converted to standard form would be 3.33 times 10 to the negative 10th power so this becomes 3.33 times 10 to the negative 10th and that would be our answer for this one. Now let's look briefly at how we can do calculations with these using a scientific calculator and we give an example here of a website that you can use so this is an online calculator that you can use it does not depend on what kind of system you are running you can go to the website linked to here and that will open up this calculator and allow you to use it. Now how do we go about using a scientific calculator for these types of calculations? In reality it depends on the specific calculator everyone is different on many calculators there is a key label double E or maybe EXP and there are multiple variations of those and other things that can be used as well but the key thing is that you want to use this scientific notation key here you do not want to use the multiplication key so even though we write out a number as 2.5 times 10 to the third you do not use a multiplication key when you enter the number when you get to this you would use the EXP key so what we would use on the calculator here to see is you could enter a number here 2.3 and then you would use the double E key in order to enter that so 2.3 times 10 to the 16th minus 9.0 times 10 to the 15th you are using this double E key instead of the multiplication key and then you just enter your number another example 6.0 times 10 to the 18th gets written as 6.0 E18 and that E again comes from this double E key the other numbers you would enter exactly as normal but when we write it this way or see it this way in the calculator then a 6.0 this means this part means for example 6.0 times 10 to the 18th that is simply the representation in the calculator and it is that double E key is what you want to use you never want to use the multiplication key when you are entering scientific numbers in scientific notation into a calculator so let's finish and summarize here with what we have gone over today and first of all we talked about scientific notation and reviewed that a little bit it is a way to more conveniently express the very large and very small numbers that we use in science classes we work separately with the digits and exponential portions if we are doing these calculations by hand you have to separate out the two and do some work there combine them and then take them apart do the calculations and then combine them back together if we use a scientific calculator however this can be done in a single step so that concludes our discussion of calculations using scientific notation we will be back again next time for another topic in math help for science courses so until then have a great day everyone and I will see you in class