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# Scientific notation examples

## Video transcript

it always helps me to see a lot of examples of something so I figured it wouldn't hurt to do more scientific notation example so I'm just going to write a bunch of numbers and then write them in scientific notation and hopefully this will cover almost every case you'll ever see and then at the end of this video we'll actually do some computation with them to just make sure that we can do computation with scientific notation let me just write down a bunch of numbers and say I have zero point zero zero eight five two that's my first number my second number is seven zero one two zero zero zero zero zero zero zero zero zero I'm just arbitrarily stopping the zeroes the next number is zero point zero zero zero zero zero zero zero zero I'll just draw a couple more I have to keep saying zero that you might find that annoying five zero zero the next number right here there's a decimal right there the next number I'm going to do is the number seven two three the next number I'll do I'm having a lot of seven zeroes to point six and it's zero point six and then let's just do one more just for just to make sure we've covered all of our bases let's say we do eight to three and then let's throw some an arbitrary number of zeroes there so this first one right here what we do if we want to write in scientific notation we want to figure out the largest exponent of ten that fits into it so we go to its first nonzero term which is that right there we count how many positions to the right of the decimal point we have including that term so we have one two three so it's going to be equal to this so it's going to be equal to eight that's that guy right there 0.5 two so everything after that first term is going to be behind the decimal so 0.5 two times 10 to the number of terms we have 1 2 3 10 to the minus 3 another way to think of it this is a little bit more this is like 8 and 1/2 thousands right each of these thousand so we have eight and a half of them let's do this one let's see how many zeros we have we have three six nine twelve so we want to do again we start with our largest term that we have our largest nonzero term in this case this is going to be the term all with the left that's our seven so there's going to be 7.0 one two it's going to be equal to seven point zero one two times ten to the what well it's going to be times ten to the one with this many zeros so how many things we we had a one here and then we had one two three four five six seven eight nine ten eleven twelve zeros now I want to be very clear you're not just counting the zeros you're counting everything after you're counting everything after this first term right there so it'd be going to a one followed by twelve zeros so it's times ten to the twelfth just like that not too difficult let's do this one right here so we thought we go behind our decimal point we find the first nonzero number that's our five so it's going to be equal to five there's nothing to the right of it so it's five point zero zero if we wanted to add some precision to it but it's five times and then how many numbers to the right or behind to the right of the decimal do we have we have one two three four five six seven eight nine 10 11 12 13 and we have to include this one 14 so 5 times 10 to the minus 14th power now this number you might might be a little overkill to write this in scientific notation but it never hurts to get the practice so what's the largest 10 that goes into this well 100 will go into this and you could you could figure out 100 or 10 squared by saying okay this is our largest term and then we have to you can say zeros behind it because we could say 100 will go into 723 so this is going to be equal to seven point two three times we could say times 100 but we want to stay in scientific notation so we'll write times 10 squared now we have this character right here what's our first nonzero term it's that one right there so it's going to be six times and then how many terms we have to the right of the decimal we have only one so times ten to the minus one and that makes a lot of sense because that's essentially equal to 6 divided by 10 because 10 to the minus 1 is 1 over 10 which is 0.6 one more so let me throw some commas here just to make this a little bit more a little easier to look at so let's take our largest value right there we have our 8 so it's going to be write a little neater this is going to be eight point two three eight point two three we don't have to add the other stuff because everything else is zero times 10 to though well we just count how many terms are after those 8 so we have one two three four five six seven eight nine ten it's eight point two three times ten to the ten I think you get the idea now it's pretty straightforward and more than just being able to calculate this which is a good skill by itself I want you to understand why this is the case in hopefully that last video explained it and if it doesn't just just multiply this out literally multiply out eight point two three times ten to the ten and you will get this number maybe you could try it with something smaller than ten to the ten maybe ten to the fifth and once you get a different number but you'll have you'll end up with five digits after the eight but anyway let me do a couple of more computation examples let's say we had let's say we had the numbers point let me just make something really small six four and let me make a large number let's say I have that number and I want to multiply it I want to multiply it by let's say I have a really large number three two I'm just going to throw a bunch of zeros here don't know when I'm going to stop let's say I stop there so this one you could multiply out but it's you know it's you have to count well it's a little difficult but let's put it in a scientific notation one it'll be easier to represent these numbers and the hopefully you'll see that the the multiplication actually gets simplified as well so this top guy right here what how can we write him in scientific notation it would be six point four six point four times ten to the what one two three four five six have to include the six so times 10 to the minus six and what can this one be written as this one is going to be three point two right three point two and then you count how many digits are after the three one two three four five six seven eight nine ten eleven so 3.2 times 10 to the 11 so if we multiply these two things this is equivalent to six let me do it in a different color this is equivalent to six point four times ten to the minus six x times three point two times ten to the eleventh which we saw in the last video is equivalent to six point four times three point two I'm just changing the order of our multiplication times ten to the minus six times ten to the eleventh power and now what will this be equal to well to do this I don't want to use a calculator so let's actually let's just calculate it so six point four times three point two let's ignore the decimals for a second we'll worry about that at the end so 2 times 4 is 8 2 times 6 is 12 nowhere to carry the 1 so it's just 128 put a zero down there 3 times 4 is 12 carry the 1 3 times 6 is 18 we got a 1 there so it's 192 all right yeah 192 you add them up you get 8 or 1 plus 9 is 10 carry the 1 you get 2 now we just have to count the numbers behind the decimal point we have one number there we have another number there we have two numbers behind the decimal point so you count one two so six point four times three point two is equal to 20 point four eight so this is going to be 20 point four eight times 10 to that we have the same base here so we can just add the exponents so what's minus six plus 11 so that's 10 to the fifth power right right minus 6 and 11 10 to the fifth power and so the next question might say I'm done I've written I've done the computation and you have and this is a valid answer but the next question is is this in scientific notation and if you wanted to be a real stickler about it it's not in scientific notation because we have something here that could maybe be simplified a little bit we could write this let me do it this way let me divide this by ten so any number we can multiply and divide by ten so we could rewrite it this way we could write one over ten on this side and then we can multiply times ten on that side right that shouldn't change the number if you divide by 10 and multiply by 10 that's just like multiplying by one or dividing by one so if you divide this side by ten you get 2.0 for eight and you multiply that side by ten you get two times ten to the times ten is just times ten to the first you can just add the exponents times 10 to the six and now if you're a stickler about it this is good scientific notation right there right there now I've done a lot of multiplication let's do some let's do some division let's do some division let's divide this guy by that guy so if we divide if we have 3.2 times 10 to the 11th power divided by six point four times ten to the minus six what is this equal to well this is equal to three point two two over six point four we can just separate them out because it's associative so we could we could say this is its this times 10 to the 11th over 10 to the minus six right if you multiply these two things you'll get that right there so three point two over six point four this is just equal to 0.5 right 32 is half of 64 or 3.2 is half of 6.4 so this is 0.5 right there and what is this this is 10 to the 11th over 10 to the minus 6 so when you have something in the denominator you could write it this way this is equivalent to 10 to the 11th over 10 to the minus 6 is equal to 10 to the 11th this is equal times 10 to the minus 6 to the minus 1 or this is equal to 10 to the 11th times 10 to the 6th and how what did I do just there this is 1 over 10 to the minus 6 so one over something is just that something to the negative one power and then I multiply the exponents you can think of it that way and so this would be equal to 10 to the 17th power 10 to the 17th power or another way to think about it is if you have one you have the same bases 10 in this case and you're dividing the and you're dividing them you just take the one in the numerator and you subtract to the exponent the denominator so it's 11 minus minus 6 which is 11 plus 6 which is equal to 17 so this division problem ended up being equal to 0.5 0.5 times 10 to the 17th which is the correct answer but if you wanted to be a stickler and put into scientific notation we want something maybe greater than 1 right here so the way we can do that let's multiply by 10 on this side let's multiply by 10 on this side and divide by 10 on this side or multiply by 1 over 10 remember we're not changing the number if you multiply by 10 and divide by 10 we're just doing it to different parts of the product so this side is going to become 5 do it in pink this side is going to become 5 right 10 times 0.5 is 5 times 10 to the 17th divided by 10 that's the same thing as 10 to the 17th times 10 to the minus 1 right that's 10 to the minus 1 so it's equal to 10 to the 16th power 10 to the 16th power which is the answer when you divide these two guys right there so hopefully these examples have given filled in all of the Glac gaps or the uncertain scenarios dealing with scientific notation if I haven't covered something feel free to write a comment on this video or pop me an email