in this video we're going to review scientific notation so what is it and why do we use it scientific notation is a simple way to write very large numbers or very small numbers for example instead of writing the number 1 billion which has a lot of zeros you could simply represent that value by writing 1 * 10 9 instead of writing let's say 0.0000 64 you could simply write 6.4 * 10- 6 so scientific notation is very useful for representing very large or small numbers so you need to be able to convert a number from scientific notation into decimal notation or standard notation let's start with examples relating to that topic for example 4.5 * 10 1 how can we convert this number from scientific notation to decimal notation 10 the 1st power is simply 10 4.5 * 10 is 45 now what about these examples 2.3 * 10^ 2 and 7.4 * 10 3r go ahead and write these values in decimal notation 10 10^ 2 is basically 100 so it's 2.3 * 100 which is 230 7.4 * 10 3 is 7.4 * 1,000 10 3r is basically a one with three zeros if you multiply 7.4 by 1,000 you get 7,400 another way in which you can do this is you can move the decimal three units to the right so for example if I have 4.28 time 10 4 I just need to move the decimal point four units to the right so I have a four it was here that's one space two three let's add a zero and then four so the decimal point is now here it's equal to 42,800 try these examples 2.7 * 10 5 8.36 * 10 7 4154 time 10 to the let's say 9 and 3165 * 10 3 so if it's 10 the 5 you need to move the decimal five units to the right so first let's write a two a seven and then let's move it so this is one and then this is the second place so let's put a zero there and then three and then four and then five so here it is that's 270,000 so let's try the next example so the decimals here 1 2 3 4 5 6 7 so basically that's 83 M 600,000 now for the next example it's 10^ the 9 so here's the decimal 1 2 3 4 5 6 7 8 9 so it's 4 billi1 154 million now the last example straightforward we just got to move it three units to the right so it's going to be after the five so it's 3,165 so as you can see whenever the exponent is positive the number is very very large but what about when the exponent is negative let's work on some of those examples let's say if we have 3.4 * 10us 2 what is that equal to in decimal notation so when you have a negative number it's going to be less than one it's between zero and one overall it's still positive but it's just very small so first let's write 3.4 this time we're going to move the decimal two units to the left so this is going to be one and then two so this is equal to 0.34 let's try some more examples try these two 4.5 * 10us 3 and 3.74 6 * 10 Theus 4 so first let's write four and five the decimals here this is one two three so now the decimal is here so this is equal to 0.0045 now for the next example we need to move four spaces to the left 1 2 3 4 so this is going to be 0.0000 3746 so now it's your turn try these problems 2.47 * 10us 3 4.61 * 10 - 5 and 2973 * 10- 8 so let's start with this problem 1 2 3 so this is 0.247 now what about the next one so we got to move five spaces to the left one 2 3 4 five so this is 0.00 0461 now for the last one we got to move eight units to the left so this is one two 3 4 5 6 7 8 so it's 0.00000000 looks like we have a total of 7 Zer and then 2973 so initially we were here and we traveled eight units to the left it's always good to double check but this is the answer now let's work backwards let's say if we have a number in decimal notation how can we convert it to scientific notation so this is 4680 the decimal point is here so if you recall large numbers are associated with positive exponents small numbers are associated with negative exponents so for this problem we should have a positive exponent our goal is to move the decimal between the first two numbers that is between four and six 1 2 3 so we got to move it three units or three spaces to the left so it's going to be 4.68 * 10 3 now you really don't need the zero you could add it if you want to but we can leave the answer as 4 .68 * 10 3 so let's try some more examples what about 32,500 and 476 convert these numbers to scientific notation so in this example we got to move it one 2 3 four spaces to the left so it's 3.25 * 10^ 4 and for this one we need to move it two units to the left so this is 4.76 * 10 2 try these 37,000 950 450,000 and 38 million convert it to scientific notation so the decimals here we got to move one 2 3 four units to the left so it's 3.79 5 * 10 4 for the next one we got to move 1 2 3 four five spaces to the left so it's 4.5 * 10 5 and for this one we going to move 1 2 3 4 5 six 7 units to the left so it's 3.8 * 10 7 now let's try some examples with very small numbers let's say if we have 024 we need to move the decimal point between the first two nonzero numbers so we got to move it two units to the right so therefore it's going to be 2.4 * 10us 2 now what about 0.36 and 00 071 try those so for this we got to move it 1 2 three four so this is 3 point actually not four units only three I take that back so 1 2 3 we need it between the three and the six so this is 3.6 * 10us 3 and for this one 1 2 3 4 units to the right so it's 7.1 * 104 now I want to highlight something that's very important so let's compare these two problems 086 and 4254 right now each of these numbers are multiplied by 10 0 because 10 0 is 1 anything raised to the 0 power is 1 when we move the decimal point to the right the exponent will decrease it decreases by two so this becomes 8.6 * 10us 2 when we move the decimal point let's say to the left the exponent increases in this case by 3 so this is 10 3r so make sure you know this anytime you move the decimal point to the right the exponent is going to decrease if you move the decimal point to the left the exponent will increase for example let's say if you have the number 42.8 * 10 5 this number is not in proper scientific notation because in scientific notation 42.8 that that number shouldn't be more than 10 it should be between 1 and 10 so we got to move the decimal point one unit to the left the decimal point should be between the first two numbers in order for it to be in proper scientific notation so if we move the decimal point one unit to the left what's going to happen to the five should we increase it to six or decrease it to four anytime you move the decimal points to the left the exponent will increase so we should increase it to six so this is 4.28 * 10 6 now if you wish to prove it here's what you can do 42.8 is basically 4.28 * 10 you can prove that with a calculator and 10 1st * 10 5th is 10 6 if you recall from algebra whenever you multiply uh two common bases you can add the exponents 2 + 3 is 5 so 1 + 5 is 6 and that's how we can get this answer convert the following numbers into proper scientific notation so let's try 37.4 * 10 5 426.7 0046 * 10 the -3 0 81 * 10- 4 and let's say 76.1 * 105 so let's remember the general rule whenever you move the decimal point to the left the exponent will increase and if you move the decimal point to the right the exponent should decrease so always keep that in mind in the first example we're going to move it to left that is one space to the left so therefore it should go up so it's going to be 3.74 * 10 6 now let's try the next one so this time we got to move it two units to the left so the exponent should increase by two so it's 4268 * 10 9 now for the third example we need to move the decimal three units to the right so therefore we need to increase not increase rather uh decrease the exponent by 3 so it's going to be 4.6 * 10- 6 for the next one we got to move it two units to the right so it's going to decrease by two -4 minus 2 is -6 so it's going to be 8.1 * 10 -6 and then for this example we got to move one unit to the left so it's going to increase by one so it's going to be 7.61 * 105 + 1 is4 so anytime you mooving to the left increase the exponent by the number of spaces that you move to the left and if you're going to move to the right uh decrease it now let's go over a mixed review convert the following numbers into decimal notation let's see if you remember everything so let's say if we have 42,900 0476 3,856 and 0024 so in the first example we have a large number so the exponent should be positive so we got to move the decimal four units to the left as you move it to the left it's going to increase four the exponent is going to increase by four starting from zero so it's going to be 4.29 * 10 4 As you move it to the left the exponent goes up here we're going to move it to the right particularly two units to the right so starting from zero the exponent will go down by 2 so it's going to be -2 so it's 4.76 * 10 -2 As you move the decimal to the right the exponent will decrease starting from zero for this example we need to move three units to the left so it's going to increase by three relative to zero so it's going to be 3.85 * 10 POS 3 and if we move it to the right it's going to decrease starting from zero so it's going to be uh ne4 it's going to go down by four units so 2.4 * 10us 4 so as you recall small numbers like this one are associated with negative exponents large numbers are associated with positive exponents now let's work on converting it from scientific notation to standard notation or decimal form so let's say we have 3.89 * 10 3 4.21 * 10- 4 and let's say 39.7 * 10 5 and 468 * 10us 5 so we know that a positive exponent will give us a large number so this is going to be when need move three units to the right so it's going to be 3,890 now for this example we got to move to the left negative exponents will give us a small number so 1 2 3 4 so therefore it's 0.421 that's the answer to the second problem now for the third one we need to move towards the right so 1 2 3 4 five so now that's basically 3 97 with four zeros so that's 3, 970,000 now for the next example we need to move towards the left since we have an negative exponent it's going to give us a a smaller number 1 2 3 4 5 so then that's equal to 0.468 sometimes you may get numbers that are not in scientific notation or in the appropriate form but you can still use the same principles to convert it to its decimal form now sometimes you may need to multiply two numbers in scientific notation so let's say we wish to multiply 4 * 10 3 by 2 * 10 5 how can we do so first multiply 4 by 2 4 * 2 is 8 and then multiply 10 3r by 10 5th when you multiply by a common base you need to add the exponents so 3 + 5 is 8 so the answer is 88 * 10 8 try these examples multip 3 * 10 5 by 2 * 10us 3 and try these as well let's say 3 * 10 4 by 3 * 10^ 5 and also 2.1 * 104 and then 4.2 * 10 6 so first we'll need to multiply 3 by 2 3 * 2 is 6 and then we got to multiply 10 5 * 103 so if we add 5 + -3 that's pos2 so this is the first answer now for the next one we're going to multiply 3 by 3 which is 9 and then 10 4 * 10 5 so pos4 + 5 is POS 9 so it's 9 * 10 9 now for this one let's use the calculator let's multiply 2.1 * 4.2 that's 8.82 and then we can add -4 and 6 -4 + 6 is positive2 so it's 8.82 * 10^ 2 and if you want to convert it to standard notation that's 8.82 * 100 which is 8 82 let's multiply 5 * 10 4 and 7 * 10- 8 try this one and be careful and make sure to put it in proper scientific notation so let's multiply 5 * 7 5 * 7 is 35 and 10 4 * 108 that's going to be 10-4 now this is not a final answer we need to move the decimal one unit to the left if you recall anytime you move it to the left the exponent should increase so this is going to be 3.5 * 10 to the -4 + 1 is -3 so it goes up by one and so that's it try this one what is 12 * 10 7 * 12 * 104 12 * 12 is 144 and then -7 + -4 I mean positive 7+4 rather that's positive3 now we got to move it two units to the left so it's going to go up by two so it's 1.44 * 10 to the positive 5 and here's another one what is 50 * 10 4 * 30 * 10us 3 so 50 * 30 5 * 3 is 15 you just got to add the two zeros so it's 1500 4 + -3 is POS 1 now this time we got to move it three units to the left so this is going to be 1.5 * 10 we got to add three to the exponent so 10 to 4th now let's talk about how to divide two numbers in scientific notation so let's divide 12 * 10 6 by 3 * 10- 4 so first you do 12 ID 3 which is 4 and now 10 6 / 104 what you need to do is subtract the exponents so it's going to be the top exponent minus the one on the bottom 6 - -4 is the same as 6 + 4 which is 10 so this is 4 * 10 10 which is equivalent to 40 billion so that's going to be the answer if you want to know how to quickly convert scientific notation into decimal form it helps to think of these numbers in terms of Threes for example whenever you see 10 the 3 think of a th so this is 5,000 whenever you see 10 to the 6 it's a million so this is going to be 7 million 10^ the 9 is 1 billion so this is going to be 88 billion that's a quick way just to do it mentally so for example if I have 4 * 10 4 I know 10 3 is a th000 that means 10 4 is the next level higher that's 10,000 so 4 * 10,000 is 40,000 let's say if I have 8 * 10 7 if 10 6 is a million 10 7 is 10 million so 8 * 10 million is 80 million and let's say if I have 3 * 10 10 if 10^ 9 is a billion 10^ 10 is 10 billion 3 * 10 billion is 30 billion so that's just a a fast mental technique that you can use to quickly convert a number in scientific notation to standard notation let's work on some more examples what's 28 * 10 5 / 7 * 10us 3 feel free to pause the video and try this example so 28 / 7 is 4 and 10 5th / 10us 3 that's going to be 10 5 - -3 which is8 5 + 3 is 8 so this is 4 * 10 8 now we know 10 9 is 1 billion that means 10 8 is 100 million so 4 * 100 million is going to be 400 million so that's 4 * 108 what about 36 * 10 7 / 4 * 10 the 4 so 36 / 4 is 99 and 7 - 4 is 3 so it's 9 * 10 3 and 10 3r is 1,000 so 9 * 1,000 is basically 9,000 try this one 48 * 104 / 6 * 10 -7 48 / 6 is 8 and then 10 -4 / 10 to -7 so it's going to be -4 minus -7 which is basically -4 + 7 and that's POS 3 so this is 8 * 10 3 which is equivalent to 8,000 and what about 96 * 10 to5 / 6 * 10 2 pause the video and work on that example 96 / 6 is 16 and -5 -2 is -7 so this is 10^7 now this is not in proper scientific notation so we got to move the decimal to the left and when we move it to the left the exponent is going to increase by one so this is going to be 1.6 * 10^ -6 now the next time Topic in our discussion is the addition and the subtraction of numbers in scientific notation for example let's say if we have 5 * 10 3 + 4 * 10 3 if these are the same you could simply add five and four 5 + 4 is 9 so it's going to be 9 * 10 3 which is 9,000 it's like adding 5x + 4x that's 9x they're like terms so let's say if we wish to combine 8 * 10 4 - 3 * 10 4 here we have like terms so we can simply subtract 8 and three 8 minus 3 is 5 so it's 5 * 10 4 and 10^ 4 is 10,000 because 10^ 3 is a th000 so this is going to be 5 * 10,000 which is 50,000 now sometimes you don't have like terms for example let's say if we wish to add 8 * 10 4 and 2 * 10 three you can't say 8 + 2 is 10 because these numbers don't match what we need to do is convert one into the other it's always better to convert the smaller number into the larger number so we need to make this three and change it into a four so if you recall anytime we move the exponent I I mean the decimal to the left the exponent will increase so what we need to do is move this decimal to the left so this will increase by one so it's 2 * 10 4 and this is 8 * 10 4 so now that we have like terms we can add 8 +2 8 +2 is 8.2 so the answer is 8.2 * 10 4 which is the same as 8.2 * 10,000 so that's 82,000 so now try this problem 5.1 * 10 5 + 3.2 * 10 4 so let's take the smaller number or the smaller exponent and convert it to the larger one so we got to move the decimal one unit to the left so this is going to be 32 * 10 5 and let's add that with 5.1 * 10 5 so now let's add these two numbers so 5.1 + 32 you can add a zero if you want so 0 + 2 is 2 1 + 3 is 4 5 + 0 is 5 so it's 5.42 * 10 5 if 10 6 is 1 million 10^ the 5 is 100,000 so 100,000 * 5.42 that's 542,000 now what's 6.3 * 10 6 Plus 5.9 * 10 4 so this time we need to move it two units to the left so this is going to be 059 * 10 and we're going to add two to the exponent so 10 the 6 and let's combine that with 6 3 * 10 6 so now that we have the same value the same 10^ the 6 value we can add 6.3 and 059 so 6.3 + 0 59 let's add two zeros this is going to be 9 5 3 and six so it's 6359 * 10 6 and we know 10 6 represent 1 million so this is going to be 6, 359,000 and that's it let's work on some examples involving subtraction so as always focus on a number with the smaller exponent let's make it or convert it to the one of the larger exponent so let's move the decimal one unit to the left so therefore this is going to be4 2 * 10 5 so now we got to subtract these two numbers 8.30 -42 0 - 2 is a negative number so we got to borrow a one this is going to be two and this is going to become 10 10 - 2 is 8 and then 2 - 4 is Nega so we got to borrow a one 12 - 4 is 8 7 - 0 is 7 so it's going to be 7.88 * 10 5th power and if you're not sure about your answer you can always check with the calculator if you type in 8.3 * 10 5 minus 4.2 * 10 4 it gives you 788 th000 which is the same as 7.88 * 10 5th 10 5th is 100,000 now let's try one more example 3. 6 * 10 7 - 9.1 * 10 to the 5 so let's move this decimal two units to the left so that's going to be 091 * 10 to the 7 we just got to add a zero here so this is going to be 3.60 minus 091 so 0 - 1 is negative so we need to borrow one we can't borrow a one from this zero so what we're going to do is borrow a one from 60 if we take away one from 60 this is going to change to 59 and this is going to become 10 10 - 1 is 9 9 - 9 is 0 5 - 0 is 5 and so it's going to be 3509 if you type in 3.6 - 0 91 in your calculator you get 3509 so it's 3509 * 10 7 and if 10 6 is 1 million 10^ 7 is 10 million 10 million * 3 is like 30 million so this is going to be 35 million 90,000 in standard notation now let's say if you want to find the square root of 4 4 * 10 6 how can we do so if we're given a number in scientific notation well first you want to separate into two parts the square root of 4 and the square root of 10 the 6 the square root of 4 is two to find aare root of 10 6 you need to realize that the index number is two so this is really times 10 6 / 2 if you convert it back into its uh exponential form so basically you're dividing 6 by 2 and so you get 3 so this is 2 * 10 3 which is basically 2,000 try this one go ahead and simplify 9 * 10 8 so this is going to be the square < TK of 9 time the sare < TK of 10 8 the square otk of 9 is three and the square otk of 10 to the 8 you just divide 8 by 2 and so it's 4 so it's 3 * 10^ 4 which is 30,000 try this one 25 * 10 12 the square root of 25 is 5 and the sare root of 10 12 you just divide 12 by 2 so it's 5 * 10 6 or simply 5 million now it works out the the same way for negative exponents so for example if you want to find the < TK of 36 * 10^ -6 it's going to be theun of 36 which is 6 and then divide -6 by 2 so it's 6 * 10us 3 now what is the cube root of 8 * 10 9 the cube root of 8 is 2 because 2 * 2 * 2 3 * is 8 now the N9 you got to divide it by the index number which is three 9 ID 3 is three so the answer is 2 * 10 3r which is equal to 2,000 now what about the cube root of 27 * 10 12th the cube root of 27 is three because 3 * 3 * 3 is 27 the 12 we need to divide it by 3 12 / 3 is 4 so the answer is 3 * 10 4 which is 30,000 now what is the square root of 1.6 * 10^ minus 5 how would you do that the square root of 1.6 is not a nice number however the square root of 16 is four so we need to convert the 1.6 into a 16 so we need to move the decimal one unit to the right if you recall if you move it to the right the exponent will decrease by one so this is going to be 16 * 10 and then -5 - 1 is -6 theare < TK of 16 is 4 and the square root of 10^ -6 we need to divide -6 2 and that's -3 so this is 4 * 10- 3 which is .4 try this one find the sare root of 6.4 * 10- 7 so what we need to do is get 64 let's move it to the right so the exponent has to decrease again so this is going to be 64 * 108 so now we can square root each number individually the square root of 64 is 8 and the square root of 108 is8 / 2 so this is 8 * 104 and that's equal to 0.00008 now let's work on two more examples sometimes you may need to square a number rather than taking the square root of a number so what is 4 * 10us 3 S so what we need to do is square four so this is going to be 4^ 2 * 10us 3^ 2 4^2 is 16 and when you raise one exponent to another you need to multiply for example X Cub raised to the 4th is 3 * 4 that's x 12 so we got to multiply -3 by 2 so that would be -6 so it's 16 * 106 now to put it in proper scientific notation we got to move the decimal one unit to the left as we move it to the left the exponent should increase by one so this is 1.6 * 10 I'm going to add 1 to -6 so that's 5 now what is 5 * 10 4th raised to the 3 power so let's distribute the three so we got to multiply 1 * 3 which is 3 so it's 5 cub and then 4 * 3 which is 12 so 10 12 5 to the 3 power is 125 so now all we need to do is put this in proper scientific notation let's move the decimal two units to the left so this is going to be 1.25 * 10 and now we get to add two to the exponent so it's going to be 10 to 14th and once again you could check it with your calculator if you type in the original problem