okay hi everybody welcome to eighth grade math this is module 1 lesson 1 the title of this module is integer exponents and scientific notation so we're going to be looking at writing expressions using exponents and this is going to build upon what you've done in the past with positive exponents we're going to span that to integers which can include negative numbers and 0 so and we're gonna apply that to scientific notation which is it's just very useful as you saw in the powers of 10 video if you've watched that one yet so let's just get going and do a little review and move ahead okay so if you had you've seen this you're adding five threes together and we saw you've you saw years ago you can write that as 5 times 3 so that's one way to use multiplication as repeated addition okay so exponential notation is similar so it's different in that I'm multiplying now so if I have 3 times 3 times 3 times 3 times 3 five times I can write that with a shortcut 3 to the fifth power so I've got three used as a factor five times and I write that as 3 to the fifth so when we I want to just diagram this so I've got this number down below three we're going to call that the base and to the fifth power so that number five is the exponent okay and the whole expression we call a power okay so that's a power three to the fifth is a power the base is 3 the exponent is 5 and it means to use the base as a factor the number whatever number the exponent is okay so let's look at some examples here's here's some examples they're not in your notes yet we're not into your classwork yet but we're going to work through these together so in the first case we've got five used as a factor it looks like five times so if I wanted to write that in exponential notation then I would write my base of five and raise it to the power of five okay since those are the same and I didn't actually mean for it to be the same let's do one more so what would five used as a factor six times B well it would be that base of five to the power of six so 5 to the sixth power would get would be an equivalent expression and a much simpler kind of shorthand way of writing it okay so now we've got nine 7s used as a factor four times an example two so I'm going to use 9/7 as my base and this time I'm going to put parentheses seas around it and put it to the fourth power and there's nine seven to the fourth power okay and now in this next expression I've got the the exponential notation already and I'm going to write that out as what that means as an expression so this one would be negative four 11s times negative four 11s times negative four 11s so I used the base as a factor three times and in in this case I use parentheses to just to show that that negative sign went with the fraction there okay so in number four I've got negative two to the sixth power so that would be this base of negative two used as a factor six times one two three four five six okay so use just a factor six times and then in the last case three point eight to the fourth three point eight is my base so I can have a decimal number for a base like that times three point eight times three point eight times three point eight okay so let's look at these examples that we have here so if you've noticed sometimes I've used parentheses so it looks like in two three and four I used parentheses and down here I used them although I didn't really need to I just wanted to not let those decimal points get get mixed up okay so that's why I used them there but let's talk about why we used parentheses in two and three and four okay so let's look at example two and talk about the use of parentheses there so we had this expression 9/7 used as a factor four times and we wanted to write that in exponential notation so I said we'll use parentheses and write it like that and that means to use nine sevens as a factor four times okay but what if I had written the expression like this without the parentheses think about what that might mean and what it would mean would be to use nine as a factor four times to make nine the base and then divide by seven or to keep seven as the denominator there because whatever is next to the exponent is what is used as the base so if I want the whole nine sevens to be the base which is what this expression is saying then I need to have the parentheses around it I can't write it like this because this means use nine as the base and then you know do the fraction so these clearly are not going to be the same product alright so parentheses are very important when a fraction is is being used as the base all right let's look at another example we'll just look at example for this one right here and this one trips people up all the time negative two to the six means use negative 2 as a factor six times or I'm gonna run out of room okay negative - okay so use negative 2 as a factor 6 times and that's and that's what that is there if if I say okay negative 2 to the 6th power what is this expression telling us to do well just like in the last example we're 9/7 was the base in this example it's saying to take the opposite and use 2 as a factor 6 times 1 2 3 4 5 6 ok so this is the exponential part and the negative sign we'll deal with later so negative 2 to the 6th is this expression and in this case these as we'll see in in a few minutes we're going to do some more example these will not be the same expression they will have a different value so if I want negative 2 to be my base like this expression if negative 2 is my base I need to use parentheses around the base or the exponent is going to tell me to work on what's right next to it okay so this would be 2 to the 6th and then take the negative of it okay and that's just how we've decided to interpret these expressions we've got to have some rules for it so ok so let's move on alright so if I have I'm going to start writing these expressions with an X just X could stand for any number right so X to the first power is is just X its X used as a factor one time so we have X X to the nth power we're going to write like this X times X times times dot dot X and we're going to show that it's n times by writing it below like that what that is something erase it so let's see what else so there's some special powers so X to the second power is also known as x squared and the reason for that would be if we wanted to find the area of a square with side length X we would do x times X same thing with X to the third power it has a special name X cubed because if we want to find the volume of a cube we could do that by multiplying x times X times X are using X as a factor three times so it would be the example would be in a cube like that alright so now I'm going to have you work on these exercises exercises one through ten and write the expression that they're asking for so I want you to rewrite these in exponential notation alright so I'd like you to pause the video and work on that now okay now go ahead and check your answers against those on the screen here and I just want to point out a couple of them that that you should have been aware of so definitely parentheses are needed here wherever there's the fractions and the negatives here here's a negative fraction definitely you're gonna need parentheses around that whereas on this expression for to the seventh I didn't need parentheses as I didn't on this one X to the one hundred and eighty fifth parentheses around the negative here and then these others you just had to put in how many times that that base was used as a factor okay so pause the video check those make sure you've got them correct before moving on okay now exercises 11 12 13 and 14 all have to do with using negative numbers as a base and I want you to go ahead and work on these on your own and come up with you think what you think the answers are and and then we'll go on and make sure to correct them together okay so go ahead and exercise is 11 to 14 you're using what you know about multiplying negative numbers to come up with some ideas about whether products will be positive or negative and so you're looking for some patterns as well and so let's see how you do with that so go ahead and pause now and and do exercises 11 through 14 alright so in these in this exercise 11 you need to use this fact that hopefully you discovered last year and and still remember the negative 1 times negative 1 is a positive 1 okay there's a lot that goes into knowing and really understanding that but negative 1 times negative 1 is positive 1 so if we look at this product every time we have negative 1 times negative 1 that whole thing becomes just a positive 1 so we've got 6 pairs like that here so when when we multiply negative 1 12 times we can always put together pairs of negative ones and those will always be positive ones so this product here this first one is positive and the second one there's negative one is used as a factor thirteen times down here and so we can pair up the negative ones up to a point and then there's going to be one negative one left over so this whole part will be positive one then we'll go ahead and multiply it by the negative one and so this whole product will be negative okay okay so an important part of this problem right here is this how do you know okay so hopefully you use some sort of explanation as to how you know either you calculated it or you thought of this idea of putting pairs of negative 1 and negative 1 together to get positive 1 and then multiplying those together to get positive 1 this idea that there's kind of one left over here you might notice 12 and 13 are both positive numbers but yet one of them had a positive result in one of them had a negative result when raising negative 1/2 to that power so what is it about 12 and 13 that makes this one positive and this one negative okay and so hopefully you maybe saw that it had to do with the fact that we could pair these up because 12 was an even number and this had one left over because 13 is an odd number okay now multiplying a bunch of ones together as easy but multiplying 95 factors of 5 together could be pretty difficult so you're asked is it necessary to do all of those calculations to determine the sign of the product why or why not okay so negative 5 to the 95th right here negative 5 to the 95th so negative 5 uses a factor 95 times once again I could think about this idea that negative 5 times a negative 5 is going to be positive 25 a negative times a negative there will give us a positive product so looking at 95 factors that means I have an odd number that means I'm going to have pairs a bunch of pairs and then times a negative 5 in the end okay so these will all be positive 25 so if I take all these positive numbers and multiply them by a negative number this product is going to be negative all right so it was not necessary for me to multiply them all out but I do know that whatever number it is it's going to be a negative product all right and then let's look at the next one I have a negative base negative 1.8 used as a factor a hundred and twenty two times this one is going to be positive because 122 is even 122 is an even number and so that means I can pair it whoops I can pair them together and get all positive products which is going to in the end turn out to be positive so we've got a positive and a negative all right let's look at number 13 fill in the blanks indicating whether the number is positive or negative if N is a positive even number then negative 55 to the N is is it positive or negative well if it's even that means I can make those pairs so this is going to be positive and if the exponent is odd that means I have one left over if I made pairs to get all those positive numbers so it's gonna make the whole product negative all right let's look at exercise 14 Josie says that negative 15 times negative 15 black dot six times is negative 15 to the sixth is she correct and how do you know all right I'm not going to write my answer I'm just gonna say it but hopefully you have written yours and are taped it in to Edie puzzle so we said back earlier that if we want negative 15 to be the base that we should write it like this negative 15 we want parentheses around it to the sixth power so the question is are these two numbers here the same do they have the same value and we'll think about it going back up to what we saw up here if we have an even number for the exponent then we're going to have a positive product so here we have negative 15 to the 6 6 is an even number so that means I can pair up these negatives are going to be positive so this whole product here is positive but the way this is written here negative 15 to the 6th this is the opposite of 15 times 15 doctor thought times 15 6 times right using 15 as a base 6 times this here is a negative number okay so this one right here is the opposite of a positive number which means it's negative so let me get my pen back that number right there is negative all right so they are not the same and Josie is not correct and that's how we know okay so that just shows you that it's really important to use those parentheses to show what you mean by your base ok so here's just a reminder of that you can write this in your notes somewhere use parentheses in expressions where the base is either negative such as negative two raised to the fourth power or is a fraction 3/4 squared or 3/4 to the second power okay so we need to use those parentheses to show what we mean by the base alright okay we've got time for one more example before you go to the exit ticket and the problem set okay so I'm going to use this this idea of half-life and this is something we'll study in eighth-grade science when we talk about elements on the periodic table that that decay okay so there's some elements that they tend to decay and so one thing that gets talked about is something called half-life and the half-life of a substance just means so if something has a half-life of a hundred years that means that half the substance decays every hundred years so whatever I start with after a hundred years I would have half of it so I want to show you how that this type of application leads to use of exponential notation okay so if I start with 500 grams after a hundred years I would I could write how much I have like this I would have one half of or one half times 500 grams after a hundred years okay so now what's going to happen in the next hundred years so after two hundred years I would have one half of the amount I had before let me use my pointer here so I had this much after a hundred years so one half of this amount here I can write it as this expression and then you get this after three hundred years I'm going to have half of what I had at 200 years so 2 x 1/2 again so do you start seeing how this looks like exponential notation so let's write those over here over on the the right-hand side ok so at 100 years I have 1/2 used as a factor once at 200 used as a factor twice so I could write that as this would equal 1/2 I'm gonna use parentheses because I have a fraction here 1/2 squared times my starting mass 1/2 squared times 500 and then at three hundred years another half life I have 1/2 to the third power times 500 grams so you can see when I get to many many years so what about what if I had after you know 1500 years I would have 1/2 used as a factor 15 times and that would my exponential notation would be a really nice shorthand okay so then after n half-lives so and being however many multiples of a hundred years every hundred years I get half the substance so after n half-lives I would have 1/2 to the N times whatever my starting mass is that would be the amount of substance that I would have okay so this exponential notation is really useful in this exponential decay we call it okay so we're going to have an example in our problem set very similar to this in lesson two so this is just kind of a intro to that okay so I I think you'll do okay so do the exit ticket make sure you correct it and then move on to the problem set and make sure you get that corrected as well we're gonna try to correct work daily that's gonna be our goal this year correct your work daily and I'll be able to see how you're doing by the answers you submit on these videos so thanks for doing that and I look forward to working with you guys okay we'll see you back here tomorrow for lesson two