Math 8 3 4 Operating with Scientific Notation

we're on Lesson Four of chapter 3 which is operating with scientific notation first we're going to divide with scientific notation then we'll multiply with scientific notation and then we'll add and subtract with scientific notation here we have two numbers in scientific notation we have 4.2 * 10 7th and we have 1.4 * 10 5th and when you do math problems with these two numbers some people find it pretty daunting because the numbers are pretty large and because it's written in format we're not really used to which is scientific notation when you come across these problems with multiplication or division or addition and subtraction there's different ways you can go about it to perform multiplication or division with numbers written in scientific notation you could perform the operation in two parts and here's what they mean by that we have 4.2 * 10 7th / 1.4 * 10 5th you can deal with the regular numbers by themselves and you can deal with the powers by themselves and then put your answer together to get your answer 4. 2 ID 1.4 is 3.0 10 7th / 10 5th as we learned in previous lesson is 10 the 2 so therefore 4.2 * 10 7th / 1.4 * 10 5th is 3.0 * 10 2 addition and subtraction take little bit more work actually when you have addition and subtraction with numbers in scientific notation you need to perform the operation with common powers of 10 and let's go over to this chart to help us little bit we know that 2.54 * 10 6 is the correct way of writing scientific notation with only number in the ones place but there are other ways of showing the same number and we're going to need to do that for these problems here see can make this number 24.5 4 Move That decimal over and since made this 10 times as big need to multiply it times 10 one less time so way to think about that is if make this number 10 times bigger need to make this 10 times smaller or drop the power one the opposite is true as well here made this 10 times smaller than this one so therefore need to multiply this times 10 one extra time to keep this being the same number so for example we have 4.2 * 10 7th minus 1.4 * 10 5th these need to be common powers of 10 so therefore this number needs to become two bigger meaning need to make this 2 power of 10 smaller or divide it by 10 two times to make this equal 7 then since these are the same we can leave this as is and then subtract the decimals to get our answer we'll go through these in little more detail with some practice problems first we're going to divide with scientific notation here we have list of the mass of planets in the solar system so going in order from closest to the Sun to farthest we have this list here in scientific notation it asks us how many times greater is the mass of Jupiter than the mass of Earth write your answer in scientific notation so whenever you see these words how many times greater we know that's division problem so basically we're asking how many Earths can fit into Jupiter which means we take Jupiter divided by Earth well Jupiter is 1.89 * 10^ the 27th power over Earth which is 5.97 * 10 to the 24th power so just like we showed here we could treat them separately with multiplication and division so since we're dividing we can treat these separate and we could treat these separate so 1.89 / 5.97 1.89 divided 5.97 that gives us an answer of 3166 we'll say now we can deal with these ones here 10 27th / 10 24th we learned in previous lesson that you subtract the powers here so that becomes 10 3r so here's our answer 03166 * 10 3r as we learned with scientific notation before this is not correct answer we need to move this over one to become like this and since we made this 10 times as big we need to make this 10 times smaller so make this one less power of 10 so if we do that we move this one over making it 3166 and that'll be times 10 now to the second power so Jupiter is this many times greater in size than the earth now we can multiply with scientific notation says the mass of the Sun is approximately 3.1 * 10 6 power times greater than the mass of Mars says find the mass of the Sun and write your answer in correct scientific notation well if we have the mass of Mars and it's that many times greater than it we need to multiply Mars times the Sun so we have 6.42 * 10 to the 23rd and that's times 3.1 * 10 6 same rules apply for multiplication and division we can multiply these by each other and we can multiply the powers of 10 by each other 6.42 * 3.1 would give us 19902 so now what power of 10 are we multiplying that by we have 10 23rd * 10 when you multiply Powers you add the exponents so that' be time 10 29th hopefully you recognize that this is not correct scientific notation we need to make this 10 times as small or move this decimal over one place so since we made this 10 times as small we need to multiply it times 10 again making this 10 to the 30th power so 1.99 02 * 10 to the 30th power and then the mass of the sun be measured in kilog now we can add and subtract in scientific notation so it says the mass of the Moon is about 7.35 * 10 22nd kilog find the approximate combined mass of Earth and its Moon write your answer in correct scientific notation so it tells us to find the combined mass and whenever we see that word combined that's going to tell you to add so we're looking for Earth and the moon here well the Earth is 5.97 * 24th and we notice that the moon does not match up with the power of 10 for this one and we said that with addition and subtraction they have to have common powers of 10 so there's two ways you can do this we can drop this down to 22 by changing this around or we could change this around and make this times 10 24th here's little hint for you it's almost always easier to make the number bigger to match this one so let's do that we have 7.35 * 10 to 22nd we want this number to go up so therefore to keep the number the same this has to go down if we divide it by 10 twice so 0.07 35 since we divided by 10 twice we can now multiply two more times time 10 so times 10 24 times instead of 22 times so here's our new number 0.07 35 * 10 2 24 so that's what we're adding here plus 0.735 * 10 24th since we're adding we don't do anything with the scientific notation part but we do add the decimals 5 3 14 add that making it 10 Dr the decimal so we have 6.04 35 and that's in correct scientific notation times 24th so that's how many kilogram combined the Earth and the moon would be now it ask us to find how much greater that means it's going to be subtraction problem is the mass of Neptune than the mass of Venus so how much greater is Neptune than Venus so we would subtract Venus from Neptune so Neptune is 1.02 * 10 26 Venus is 4.87 time 10 to the 24th so since these don't match up have to change one of them remember it's easier to make this go up than it is to go down so to make this bigger we need to make this smaller and need to get two bigger so let's move this over twice so that's like dividing it by 10 two times and since we divided it by 10 two times that means we can multiply times 10 two more times so here we have it we have 1. 02 * 10 26 power minus 0.487 * 10 26 power we're going to subtract here let's add couple zeros we' cross the two out making that one making this 10 making that nine 10 - 7 is 3 9 - 8 is 1 we can't do 1 - 4 so let's cancel this out so we have 11 minus 4 making seven and we have 9 point and zero so 0.971 3 * 10 26th we know this is not correct scientific notation we need to multiply this times 10 to make correct scientific notation which means we multiply times 10 one less time over here so we move that decimal over giving us 9713 we multiply times 10 1 last time so times 10 to the 25th