College Algebra - Lecture 3 - The powers that be-Exponents

well we've looked at numbers now we've looked at the language of mathematics and now we're going to examine the powers-that-be exponents and the first thing we'll look at are integer exponents so let's see what we've got here integer exponents well I'm going to motivate this with an example first and let's imagine that we'd like to multiply the following 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 you got that well you know that's a lot of two's to multiply it's awkward to write down and I'd like to have some kind of shorthand for that so what I will do is use this shorthand I will say I'm going to take 2 and I want to multiply it if you count there there are 10 of them I want to multiply it 10 times so I'll write 2 to the 10th that is nice and compact as opposed to multiplying out like this and just for those of you that would like to know what 2 to the 10th is that burning question 1,024 happens to be the same thing as one kilobyte if you're into computers but the important thing here is this shorthand this shorthand is that what I'm going to be describing right now so let's make this official we have a number we're multiplying at a certain number of times and I'm putting the number of times up here and we're going to want to expand that so let's go ahead and get an actual definition down definition and this will be for any real number so any real number a alright here's what I'm going to define a times a times dot dot dot up to a suppose we multiply together in of those in and these are equal so I'll even mention that n equal factors what we'll do is we'll write this notation a to the M power now let me explain what we have here this shorthand notation needs explanation this number a is usually referred to as the base it is the number you are multiplying out the number n well we can call that several things it's going to be for us right now an integer exponent that's one of the words that's used exponent it's also referred to as a power that explains the title that I used for this section it's called a power it's also for those of you that know word processing it's a superscript right it's a it's a number that is above the line of the base now remember these are integers these ends here are going to be 0 plus or minus 1 plus or minus 2 dot dot dot out to infinity now right now some of those don't make much sense it's certainly clear that if I want to multiply a six times I can write a to the sixth that's the way you say it and that makes sense but it's not clear what zero would mean or what negative numbers would mean so we'll have to get to that in a moment but there we have a definition so let's go ahead and see how this works we need to make one other minor definition because I mentioned a to the zero power and we will simply define that for convenience to be one now that will make all of our other operations as you'll see work out this however only works for a not equal to zero so to put it another way the form zero to the zero is undefined and just as division by zero was undefined that 1 over 0 form we looked at before this is also undefined now I'm not going to show you any more than that you can sort of see why this wouldn't be something that you could easily define because the zero to any power sort of takes this number towards zero on the other hand if something to the zero is one that zero above would tend to make this one and there's a conflict there which is the reason that this remains undefined now here's an example just to show you how this works two to the zero power is one right there you go two to the zero power is one you can take negative numbers negative 105 to the zero power is one etc any real number taken to the zero power except zero is going to be one now just as a minor note here because there may be some one out here there this thinking of this you have to be a little careful in context this won't be a problem but don't confuse zero power with degrees remember when you write something like 45 degrees if you've looked at angles there's that little oh up there it looks like a zero power but that's not what it is so in case that occurred to you don't make that confusion all right well that's zero that was a special case I wanted to get out of the way let's get back to numbers two powers that make a little more sense to us at least on the surface let's start out with this a to the one well that would be a because you're multiplying a by itself once so that makes sense now how would we say that well we could say like I said a to the one or we could say a to the first power there's one way to say that let me point out right now that tooth ax is the English Way of saying raise it up put it as a superscript okay let's go on a square whoops I just said something I need to explain I should say a to the to power a to the 2 power is simply a times a there are two A's there and so we can say a to thee will say second how about that second power but as you heard me say just a moment ago we also say a squared now why would we say a squared well this actually is related to a square I'll tell you why because if you take a square of side a and you figure its area its area is this a to the second power so to say a squared seems natural and that's where this comes from now most of the exponents we see will not have their own special names like a squared but a squared does and there's one more the very next one a cubed so a to the third power is a times a times a we could say a to the third power if we like we also say a cubed and again this has a geometric meaning so why would we say that we say that because if you have a cube of sides a a and a all three sides are a what is the volume of that cube the volume is a cubed or a to the 3 power so that's the reason cubed is U is used here because that's exactly the kind of figure this comes from now a squared a cubed are the only ones that have these special names these sort of historical names all the rest of them when we have a to the N power which remembers a times a times a n times so they're n factors here we usually just say a to the nth power whatever n is so 4/4 we'd say a to the fourth power 4/5 we say a to the fifth power etc they don't have special names all right as we proceed through here I'm going to give you little warnings and tips on how to deal with exponents so first now let me give you a warning this is something that we can talk about right now so let's go ahead and do this this will have bearing later too this will recur warning and what's the warning about it's about even powers and negative numbers and numbers of course I mean again real numbers so this is a warning about even powers and negative numbers and here's why let me just show you one examples you'll be sensitized to this take minus 2 a negative number raise it to the fourth power now what does that mean that means minus 2 times minus 2 times minus 2 times minus 2 well minus times a minus you may recall is a positive number the same thing with the second pair so this is 2 times 2 which is 4 times 2 is 8 times 2 is 16 so minus 2 quantity to the fourth power 16 all right this is not equal to this by the way is the not equal sign as symbols turn up in the course I'll introduce them to you that's the not equal sign this is not equal to minus 2 to the fourth written that way notice there are no parentheses here so the grouping that is intended here this is minus quantity 2 to the fourth when the minus is out front it is assumed to go over the entire expression up here the minus is only for the two because I have parentheses around that now you see what happens here minus 2 to the fourth 2 to the fourth is 16 so this is minus 16 and that certainly is not equal to 16 so be very careful I'll put this down here placement of this minus sign and grouping matters okay grouping and the placement of the minus sign my matters is running off the screen here a little bit but you heard what I said you want to be very careful how you read these and the more examples we see throughout the course the more you'll become sensitized to this and you'll have a chance later to do some of your own examples all right let's summarize where we are at this point because I did digress into the warning so far we've seen what we've seen a to the N defined for powers what M equals 0 1 2 3 etc but I said in this section we were going to talk about integer exponents and integers include negative 1 negative 2 negative 3 so but what about negative integers what could that possibly mean what could things like a to the minus 1 a to the minus 2 a to the minus 3 mean now we don't want them to have an arbitrary meaning they have to be connected to our number system somehow so we ask ourselves what could they mean well I'm going to leave you with that question now and just let you wonder about it for a little bit we'll come back to this but you might ask yourself what could that possibly mean powers that are numbers like this it's easy to see and the zero we've defined but what are these going to be so I'll leave that for you to ponder go back to my list and the next topic under integer exponents will be operations with integer exponents now that we know what they are let's go ahead and do some operations with integer exponents alright first let me show you an example first we'll do an example to kind of set the scene here imagine you have three to the fourth well we know what that means now that means three multiplied by itself four times and we want it to multiply that by three to the seventh say which is three multiplied by itself seven times well if you write this out three times three times three times three that's three to the fourth x three times three times three times three times three times three times three seven of them now you know from your long experience with numbers like this in your elementary school years that the parentheses don't matter we can just consider this a long string of threes how many threes are there there are eleven factors four plus seven so that only makes sense that three to the fourth times three to the seventh ought to be three to however many threes there are eleven of them so it looks like we might have a rule this suggests for following our rule generated from this idea now I'm doing this rule first because it's a very important rule and we'll get whatever we want from this so I'll even say that this is an important rule and based on what we just saw the rule goes like this if you have a to the M power times a to the N power then based on our previous example it's clear that what you do since you have mas multiplied in n A's multiplied you simply add the exponents so suddenly we have a rule a to the M plus n see these rules don't come out of nowhere they actually are based on examples I'm not going to prove this in general I think you see how this works from the previous example now this only works of course let me make that point at least this month D must have the same base if you were multiplying two different numbers two powers this wouldn't work but if it's the same number two powers you simply add the powers because that's how many times you're multiplying a so for example five to the 11th times five to the ninth is five to the 20th how did I get 20 I took 11 and 9 and added I noticed that the bases were the same and so I've just followed the rule and so you can try out examples like this on your own all right so now we have a simple rule and let me ask a natural question at this point question suppose we consider the following situation suppose we consider five to the tenth times five to the Box something don't know what equals one suppose we look at that now the question is what could this be what number might we put up there that would make this expression true well let's see by the rule on the previous page when you multiply two powers of the same base you add the powers so I could rewrite this as five to the ten plus box that's equal to well I'm going to rewrite one also I'll rewrite one is five to the zero because remember five to the zero is 1 so 5 to the 10 plus box equals 5 to the zero well if the two bases are the same that means these powers up here better be the same so equating powers what do we end up with 10 plus box equals zero well that's an easy equation let's solve for box subtract 10 from both sides of it so that ends up moving the 10 over and I'll move it over here we end up with box equals minus 10 ah now I'm getting an idea here so what does this mean if boxes minus 10 let me rewrite what I have up here with the box filled in thus I have that five to the tenth times 5 to the minus 10 equals one apparently I've discovered something but wait a minute but we already know something we already know that five to the tenth times 1 over 5 to the 10th equals 1 because any number times its reciprocal is equal to 1 well if five to the tenth times 5 to the minus tenth is 1 and 5 to the tenth times 1 over 5 to the 10th is 1 then this and this must be the same thing hence we decide that 5 to the minus 10th is 1 over 5 to the 10th aha now we have something that we can define because this makes sense with our number system and what we've been doing so definition a - the - and will be defined to be one over a to the N based on our example with five a moment ago now this is only going to work it's got a little bit of a restriction a can't be 0 that makes sense because we don't want to divide by 0 we already know we can't do that and the ends the only ends I'm talking about here are N equals 0 1 2 3 etc because these we've already defined powers 4 and so now we have them all defined for negative numbers so let me say something about what we've just done here this definition first of all it's a new use of the symbol bar okay that symbol is now used in a different way it is not subtraction which is exemplified perhaps by 13 minus 5 it is also not negation as in say minus 7 Minda Gatien you might remember is going left on the real line this new symbols usage is neither one of these if you wanted to think about it I suppose you could say what it is is it says rewrite this as the reciprocal 1 over a to the N so the - simply says rewrite me as a reciprocal but I'm not going to say that we're just going to memorize this rule as it stands because we know it makes sense now and let's look at a couple of examples 13 to the minus 4 power what does that mean that's 1 over 13 to the 4 I don't know what 13 to the 4 is and I don't care right now I just want to illustrate what the negative power means here's another example the number here does not have to be positive suppose I took minus 6 to the minus 3 power now these are different uses of the minus sign this is 1 over minus 6 to the 3 power so the minus here indicated reciprocal it's just a plain 3 down here now well with these two examples I begin to see something else that I want to talk about so I'll do a little by the way here notice the following well notice again because I think we talked about this briefly in another example earlier if you take minus 1 times minus 1 you get 1 now that's what that's minus 1 squared and even number also if you took minus 1 cubed minus 1 times minus 1 times minus 1 what do you get well two of them as you know from the previous case equal 1 times minus 1 equals minus 1 and so on well this certainly suggests a pattern in fact it's a pattern worth writing down so I will write it down in the form of a mathematical fact and let's call it handy lemma so I get to use that word lemma a handy lemma here's the handy lemma minus 1 to the odd n power will always be minus 1 and minus 1 to the even n power will always be 1 and what ends am i talking about let's be clear where n is 0 plus or minus 1 we've got them both defined now plus or minus 2 all the way out to infinity in other words all of Z so this turns out to be very handy if you take minus 1 to any odd power you get minus 1 if you take minus 1 to any even power it is always 1 so when you're faced with what's minus 1 to the 101 power you don't have to do any work 101 doesnot is an odd number so this is going to be minus 1 and so on so this turns out to be a surprisingly useful fact there's not much to it it seems but you'll see it again and again so let's continue on with our rules of exponents here we write down other rules because we haven't exhausted them yet and just so we have it on this page let me recall the one rule that we do have we have a to the M times a to the N equals a to the M plus n now we have that rule and this now works for all of the integers we've seen so let's go ahead and see what else we can write down and these you have to understand what these rules are these are common-sense rules if you practice with integers integer exponents at all you'll see that these things work out for example what would a to the M all taken to the N power B now you can just write the rule down but let me show you why it's true let's put an intermediate step in here before I actually write out what this is equal to the intermediate step says well what does this mean this means the thing inside which happens to be a to the M is multiplied by itself n times so this is a to the M times a to the M times dot dot times a to the M and how many of them are there there are n factors here well how many M's do you have n M's so how many numbers in all do you have you have a to the M times n and there's your rule a to the M I'll take it to the nth power ends up multiplying these two numbers but you see why that's true now so even if you were to forget this you could try it with a couple of numbers and see what the rule has to be that's the way you want to remember these rules you don't want to just commit these to memory without understanding of what's happening in the middle all right let's take another one suppose you have two numbers a times B a pair multiplied together and then taken to the nth power well before I write down the other end let me put in the middle what does this mean it means a be the quantity is taken to the nth power multiplied by itself a be a be a be n times n factors well from what you know about numbers multiplication can be reordered any way you like so I have how many A's here well I have n of these abs so I have n A's I also have n B's so if I reordered them differently this is a to the M times B to the M and there I have a rule which says the N power distributes over the two inside factors however you now see why that's true so it shouldn't be a mystery that this works and again if you know why things work then if you happen to slip up you can reconstruct them that's how people really remember most of mathematics all right let's do a couple more here what if you have different powers same base different powers a to the M over a to the N a division what would this be well here's a way to think about this this is a to the M times 1 over a to the N so that's just rewriting the bottom part is a single fraction that's what that means so this is a to the M times well 1 over a to the N we define that to be a to the minus N and now we have a situation where we have one base to a power times the same base to a power what are we supposed to do we're supposed to add the powers that was the very first rule we saw so this is a to the M minus n so this first one becomes a to the M minus n however there's another way to think about this so let me show you that other way or instead of rewriting it a to the M times 1 over a to the N let me go ahead and move the a to the M down write this 1 over a to the minus M times 1 over a to the M so this first one is rewritten in a different way remember now the minus says go to the other side it's a flip over do a reciprocal so that's why a to the M is 1 over a to the minus M well now I have these two on the bottom multiplied together when you multiply them together you add the powers as we saw the first rule so this is now a to the N minus M so it depends on what you want here do you want to have a fraction that has a 1 on the top and a number on the bottom or do you just want a number on the top and it'll depend on what you're looking for in a given case okay here's one more what if you had a fraction a over B to the N now we did this a moment ago with a times B now we're doing the division well again this is a over B times a over B dot dot dot times a over B there are n factors so how many A's are there on top a to the M how many B's on the bottom B to the N and there you have yet another rule to add to your collection so you will have time in a little bit to practice all of these rules but let me give you one more warning this is a warning and remember what we saw before I'll even write it up here before we saw that if you had minus two to the fourth that was -16 because you really thought about it as broken down into minus times the quantity two to the fourth all right here's my warning be careful be careful what object what mathematical object that is your exponents or your powers if you like that word your powers or exponents apply to now that really repeats this idea but I want to go ahead and show you a few more examples because there are some things that you can run into some difficulties you could run into if you don't watch it so let me show you a couple of standard kinds of errors that can be made look at a times b cubed now I warn you that this is not the same thing as a B quantity cubed this 3 power only goes with the B if you want it to go with the a the B you have to put parentheses because a b cubed as you know from our previous set of rules is a cubed b cubed and that certainly is not the same thing as a times b cubed so there is one thing that you want to be careful about let me show you a similar example same problem really but looks a little different a times B to the minus 1 well if you follow my advice here the same advice works here this minus one only applies to the B so this is not a B quantity to the minus 1 but you can check that more directly a times B to the minus 1 is a over B B to the minus 1 is 1 over B a B to the minus 1 is 1 over a B you see these two are not at all the same so it does matter how you write things what you group things as so keep that in mind let me show you a couple of more a couple more fallacious arguments here a cubed times a to the fourth now you know from our rule we've said it a couple of times what you're supposed to do you're supposed to add the powers this is definitely not equal to a cubed to the fourth power because as you know from the rule this is a to the seventh 3 plus 4/7 that's how many a's are multiplied and this one is a cubed taken to the fourth this of course is a to the twelfth and these of course are not at all alike so once you understand how these works it becomes harder to make mistakes we show you just one more here a to the minus 1 plus B to the minus 1 this is not the same thing as a plus B to the minus 1 we've never said anything about exponents distributing over a sum and this is a perfect example why that's not true a to the minus 1 plus B to the minus 1 is equal to 1 over a plus 1 over B that's what the minus 1 means on this side this is minus 1 applied to the whole thing so this is 1 over a plus B now you know from your knowledge of fractions that these two frack are not the same and if you have any doubts put in a couple of simple numbers just to remind yourself but they're not at all the same all right well we've seen a lot of operations on these integers now now that I've given you some warning what I'd like to do is show you how some exercises are worked out involving integer exponents because you'll have a chance to practice with these the exercises all look the same way they all start off with something like the word simplify and then there'll be an expression maybe like this X to the fifth Y to the minus 2 all over X cubed Y and for the sake of argument in case anyone wonders we'll just assume so we don't have any division by zero that X is not 0 and Y is not zero okay so there's no division by zero problem now our goal here is to simplify this why does anyone want to do this in algebra at this point well to give you some practice in dealing with exponents many of the operations and the the kind of functions we see later will involve exponents and you need to simplify them because frankly this is too hard to look at there's too much complication there I can't tell what's going on it's important to us to have things simple now I'm going to write down one solution why did I write one well because this is not the only way to do this the art of simplification is simply to get this to be simple everybody does it a little differently I'll show you some standard practice and you might want to imitate that but when you get to be more familiar with this you'll choose other paths so let me rewrite that here X to the fifth Y to the -2 over X cubed Y first rule of thumb and I will mention this as a tip at the end first rule of thumb is get rid of those negative powers rewrite them so that they're positive powers now what does that mean that means Y to the minus two must be re-written as 1 over Y squared so basically that's going to move that downward so let me do that first X to the fifth hours left alone on the bottom I have X cubed Y times y squared so this has been moved downward well now you can see what's to be done next we could do two jobs at once Y times y squared there's three Y's multiply that's Y cubed and then we had a rule that said X to the fifth over X cubed this is X to the five minus three and that makes sense when you think about it you have x times X times X up here times X times X three of those over X cubed equals the number one so this is the rule but that was the reason and so you end up finally with let's slide it in here x squared over Y cubed now that is much much better than the original here we have a single thing on the top and the bottom and no negative powers so the dreaded negative powers have disappeared so that's a simple example let's do another one because you need to see more of these and do do a lot of these on your own this is not what this course is about this is simply you might think of it as finger exercises and music you have to get used to this so I'll do enough of these to give you some practice X to the minus two minus y to the minus two over X to the minus one plus y to the minus one and again so that we don't have division by zero I will assume that X is not 0 and Y is not 0 all right here I go one solution my pathway to simplification first of all I'm going to rewrite it I have 4 negative powers don't like that at all I'm going to rewrite that get rid of them right away so what do I end up when I rewrite that we'll end up with I have 1 over x squared minus 1 over Y squared now that is on top and on the bottom I have 1 over X plus 1 over Y so it looks like I've made things worse well only temporarily I have no more negative exponents so I consider that a bonus all right now what I need to do on the top and the bottom because I want apply some of the rules we have for fractions I'm going to combine the top into a single fraction combine the bottom into a single fraction how do I do that well I have to get the same denominator for both of the things on the top and for both of the things on the bottom the technique for that is this is the top now and this is the bottom one over x squared well if I want my denominator to be x squared times y squared x squared needs a y squared down here well I can't change things so I multiply 1 over x squared by Y squared over Y squared that's multiplying by 1 so I haven't changed anything - do the same trick here 1 over Y squared I will multiply top and bottom by x squared over x squared again that's 1 no change but now you notice the denominators are both x squared times y squared same thing down here 1 over X I want x times y to be my denominator so I'll multiply by Y over Y to make this have the correct denominator and then 1 over Y will multiply by x over X so I have the correct denominator here so now I'm all set to write the top the top and the bottom as a single fraction each so let me do that I will bring this back so you'll be able to see what I did this is equals continue from the previous page you want to have a string of equal signs now now what do I have I have the fraction on top which will have as its denominator x squared Y squared and the top will be Y squared minus x squared on the bottom the denominator we got was X Y and on the top it'll be y plus X since you don't have the preview seat in front of you let me bring it back there is what I had so you see x squared Y squared is the tops denominator and what's on the top of that Y squared minus x squared and on the bottom X Y is the denominator for both and what's on the top y plus X and that's exactly what I have written down here well now I have a single fraction over a single fraction I know how to deal with that what was the rule let's see I copy the top so I have y squared minus x squared over x squared y squared just copy it and what's on the bottom what am I supposed to do with this I'm is supposed to flip over and multiply in other words write that as a reciprocal so let me do that multiply and flip it over so it's now XY over y plus X so I'm on my way to getting a simpler expression almost there now I'm going to do something that you probably don't expect if you don't remember this equals I had two things I'm going to make them into a single fraction let me remind you now what I had that's what I had since these are multiplied to multiply fractions remember I multiply the top I multiply the bottom so the bottom becomes x squared y squared times y plus X the top becomes Y squared minus x squared times X Y now this happens to be an expression we'll talk about later but that can be rewritten this can be what's called factored I can write this expression as the product of two other expressions we'll talk about that later but let me go ahead and do it here I can rewrite this as y minus x times y plus X so this becomes factored just like this and then still I have the XY and on the bottom I have the x squared Y squared and the y plus X and now I'm ready to finish y plus x y plus x y plus x over y plus X is 1 so I don't have to write these anymore also I have x and y here and I can pull out an x and a y here XY over X Y is 1 what's left Y minus X from here and then One X and one Y from here and I'm done this is much simpler let me remind you of what the original one looked like the original one looked like this and rewritten look like this we've improved things considerably we have y minus x over X Y very simple and straightforward and when you do further calculations with this you'll be much happier to have it this way so let me give you some tips now and then we'll take a pause and you'll be able to practice this on your own so here are some tips some of them I've already said but let's get them written down here one let's be careful with groupings if you have any doubt put in parentheses where you think there should be parentheses it's better to have them in there than not to have them in there secondly I recommend that you keep the division form the quotient form where the top is directly over the bottom in a vertical manner keep that form not another form that you will see in textbooks that looks like that the reason you don't want to do this is because this is for textbooks or you say this is also useful for calculators and computer computers and computer systems that's the way you very often have to enter things why because the computers in the calculators like to have everything on a single line but when you work it by hand this is by far superior it keeps you mentally aware of what is happening and makes the operations easy to remember so do things this way finally as another tip always rewrite negative exponents immediately negative powers immediately okay that's very important as you saw it makes things easier to work with let me give you one more example of that before I leave you these are the tips that you really want to jot down in your notes and keep aware of these are rules of thumb these are the result of experience and one more thing 1 over X cubed is more useful in the practice of writing down calculations than X to the minus 3 similarly X to the fifth now you'll believe this one I know is more useful then 1 over X to the minus fifth now these are equivalent as are the first two but if you keep things without negative powers it's going to be easier for you to follow the rules and to simplify your expressions so on that note let's take a pause and you can go ahead and try some of these on your own now that I hope you've had some experience with integer exponents we're going to talk about another topic and at first it may seem like it's not related the topic is square roots and I've got semicolon they're a pair of equal factors now what do roots have to do with exponents it's a good question and that's why I've set aside square roots first and then we'll look at other kinds of roots later so let's first talk about square roots to lead us into this all right actually there are both both of those words need explanation square and roots we haven't talked about roots at all here's what we say we say a to the second power remember we say a squared is a and sometimes we write the word perfect I'll put it in parenthesis here is a perfect square and what we say as I said is we say a squared now why again was that remember that's because if you have a square of side a then the area of that squared the area is equal to a times a a squared all right once we say that and we say also the following we say that a the number here that is taken to the second power of the base the thing being squared we say that a is its square root now the use of the word root is an older piece of terminology that has lingered on it is simply the base if you like or the number that is squared in this case so that's where square root comes from so let me say it another way so a is the square root of a squared because I'm repeating myself but I want to get this clear because a times a equals a squared that's what a root does and notice that these are a pair of equal factors factors you remember are things which are multiplied all right so square this is a pair of equal factors so now we're going to generalize and how am I going to generalize well I'm going to not start out with a number that I write as a squared I'm just going to call it X and then look for the number that would multiply by itself to give me that X so this will be in the form of a theorem I'll call it a fact just for short but it is a theorem fact and here's here is the generalization I mentioned for any X greater than or equal to zero so any real number that is either 0 or positive there exists a positive number okay a positive number whose square is X meaning I'm going to write this out so you know what I'm talking about meaning X is equal to this thing squared okay this theorem says that if you give me any number which is 0 or positive then I can find another number whose square will equal X this thing here this thing that will go in the box this is called the principal square root of X it is the positive number positive is what principal means here positive number such that when you square it you get X all right and it's denoted as follows it's denoted by this square root symbol this number is the number which one's squared and put into that box equals x now this is a funny notation in the part on the language of mathematics we talked about this notation and that it might have resulted from a corruption of drawing the letter R here you also note that I put a tail on the end that's something that I will continue to do but this is now the symbol for the number whose which when squared equals x let me write that out so you're clear on what I'm talking about meaning by definition if you take the square root of x and you square it you get X that's what the symbol means now let me notice something else while we're here notice that if you take minus square root of x and you square it you also get X so that happens too that's why the other one was called the principal square root so let me summarize the square root of x is the principal square root principle means positive here all right positive there may be other square roots in this case negative square root of x is the other square root and this of course is negative so the principal one is the positive one the other one which is called the negative square root all right let's look at some numerical examples example what is the principal square root principle square root a long time to write that out square root of 2 well the answer is using the symbolism square root of 2 I can't do any better than that because square root of 2 as you know is an irrational number that is the symbol I'm going to use and that's all I can say at this point what is the principal square root of 9 answer 3 how do I know that because it's positive and 3 times 3 is 9 what's the principal square root of 1 to start with some of the ending cases the cases that are sort of on the edge answer 1 one is its own square root because one squared equals one what is the principal square root of zero answer zero another number that's equal to its own square root Y etc okay now in general finding a square root is simple as writing down the symbol if you want to simplify this any further it's going to depend on what the number is if the number is 9 then the square root is 3 that's easy if it's 2 and you get an irrational number there's nothing you can do but write this down if you were really interested in what this number was you might get an approximation and you can do this with calculators these days but let me say the following square root finding how's that for a phrase square root finding is just here's another way to think about it on multiplying x on multiplying X into two equal factors and you can do that as the theorem set with any positive number or zero and justice repeat myself here by example two is equal to the square root of two times the square root of two I've taken two and unmultiplied it will later call that factoring into two numbers which are the same square root of two 9 is 3 times 3 etc so that's what square root finding is alright now I started this section saying what the square root have to do with exponents this is a whole section on exponents so let's pose a question here question could we write okay could we write the admittedly strange symbol okay because it's an older symbol st. simple square root of x as perhaps a power of X or an exponent of X could we possibly do that well if we could this would be handy because we have a lot of rules for powers and if the same rules apply then we'd be in good shape let's see we have to make an argument we have to come up with a reason for rewriting this as a power so let's see if we can come up with one here let's see since the square root of x times the square root of x equals x that's the definition of square root after all what do we need we need X to some power unknown X to the box times the same thing X to the box equaling X okay that's what we want to try and have happen well we have a rule which says we can rewrite this so by the rule this is the same base we can write this as X to the box plus box equals x but box plus box is just like a number that's two times box so this is X to the 2 times box equals x and now this x over on the right I'm going to write it as X to the 1 power to show that it has an exponent well I have two bases that are equal I want to equate the powers or I can't equate the powers what do I have I have two times box equals one well what does box have to be then divide both sides by two box must equal one half aha I think I have an idea now we haven't had an exponent there was a fraction but now it looks like this is the right thing to do so let's see how this would apply we then can write the square root of x equals x to the one-half power see my little 1/2 there so I've taken this rather odd square root symbol and written it in a power form I'm familiar with powers we've never had rational ones before but now I have a meaning for rational X to the one-half means the square root so this on the left is an old notation and this on the right is clearly it's a rational exponent it's a power that is a rational number and this is a new and this is more useful so let's put that down this is a new more useful notation see we don't invent new notations just to be different we want to have something that makes things easier so to summarize this thus if you take the square root of x squared that is now the same as X to the 1/2 squared and what was the rule for powers that were like this x2 something all taken to another power we multiplied the powers right so this is the X to the 2 over 2 but what's 2 over 2 1 and what's X to the 1 X so that just confirms that what we have here is a new notation that does the same thing as the square root symbol did but with the new notation we have all these rules that we can now follow so let's go ahead and look at the the next topic which will generalize this so let's go back to my list of topics we looked at square roots which is a pair of equal factors and now let's go ahead and look at nth roots and rational exponents the rational exponents we've already talked about now we're going to talk about nth roots those will not give a pair of equal factors factors they will give n equal factors as you will see in just a moment so n throughs and rational exponents so this is the title of the section I am now looking at and I'm going to generalize all of the work I just did with square roots alright let's recall what I just did with square roots as I change the color on you here a bit the square root of x and remember square root is another word for the second root of x the old Doulton was the square root symbol and we have now got a new notation X to the 1/2 now let me point out one thing about the square root notation you'll see in later notations that there's a number that appears there the square root is the only one that doesn't have a number you can imagine that there's a 2 there if you like that will make the rest of the notation look similar so if square root of x is this square being second and we have 1 over 2 I ought to be able to guess that the cube root of x now cube root is the third root would be similar symbol except I put a 3 in there so that would be cube root of x and that you might expect is X to the one-third so based on what I did before I'm now generalizing to the case where I have written as a not a pair but a triple of three numbers that are the same so let's go ahead and generalize these ideas that we've just developed generalize because once you've seen it once the rest of it is very similar definition the principle instead of square root which is second root I will say principle and throat the principal nth root of a number say a is denoted the nth root of a so it looks like the square root symbol with an N here and the only symbol that you don't use a number for is square root that's where the number would be to but it's not traditionally written there so what does this mean this means that a is equal as I said earlier a is equal to the product of n equal factors so I am unmultiplied a into n factors all of which are the nth root of a so it's n equal factors each called the nth root of a so that is the fundamental idea here once again I'm going to write down what this means so you have a good sense of it meaning if you take the nth root of a and you take it to the nth power you get a now there's a little proviso here that we need to take into account this will lead us to a later part of the course so it's nice to put this in now provided that the following situation holds when n is even so we're talking about and even power here or an even root when n is an even number a must be greater than or equal to 0 it must be positive or 0 and when n is odd there is no such restriction so even has this restriction odd a is any real number so why would there be such a restriction let me go ahead and tell you why and on the next sheet here the problem is the following so the form which looks like this square root of something to the something power the form where this is even this is an even n and this is a negative number any negative number whatsoever that form is the one that I've just said is not defined in the real numbers and that is the problem it is not a real number and why is it not a real number well well it'll turn out to be a complex number so when I finish this section and we move on a bit we'll talk about numbers again and we'll come back to complex numbers but such a situation an even root of a negative number leads to what are called complex numbers they are not part of the real number system right now and but they will be later so just bear with me and accept the fact that you can't take roots of that sort so for example of roots that you cannot take like that you cannot take the square root of -1 which will turn out to be key for defining the set of complex numbers you cannot take likewise the 4th root of minus 110 etc any negative number with an even root is not a real number so if we get that out of our system we're fine oh and one small note while I'm here note the nth root of 0 is equal to 0 for all N equal 1 2 3 4 etc well I guess we didn't really define a 1st root so let's just say N equals 2 on up ok so let me recap what we've got remember by definition what is the nth root of a you want to be very clear on this it is the principle that is to say the positive and root of a now for those of you that are interested you know that square root there were two different roots there was the principle which was the positive square root and then the negative of that was the other square root so a square root was there are two square roots for a given number you might expect that there are three cube roots 4/4 roots and in general and nth roots and if you did suspect that you would be absolutely correct unfortunately in most cases the numbers are not real numbers they are complex numbers so although there are other roots will have to put off discussing some of them however we can say the following fact again fact is it is a theorem suppose you take the nth root of a to the N okay so you have an nth power you take the nth root and let's assume that a is real a real number of some kind then there are two possibilities here if you take the nth root of a you will get back a if n is odd however you will not if n is even the other case here you will not get payback and the problem is that if a is a negative number for example and you take it to an even power a negative number like minus two taken to the fourth power as you saw earlier in my example is a plus sixteen then if we go ahead and take the fourth root of the plus sixteen principle positive nth root we get four but the original number under here was not was sorry the nth root would be to the original number however was minus two I'll show you that in an example but the end result is that you get the absolute value of a you get a made positive if n is an even number so let's look at two examples so that you'll see what it is that I just tried to explain if I take the fifth root of minus 7 to the fifth power I get back minus seven because when minus seventh taken to an odd power the negative sign is not lost when I take the fifth root the negative sign is not lost however in the case of the eighth root of minus 7 to the eighth what happens the process in here minus 7 to the 8th removes the negative sign and then when I take the eighth root the principal square root the positive square root I don't realize that there's a negative sign there it's been lost so I would just write down 7 so the way we've written that in our previous theorem is that the minus 7 is its absolute value when we take the eighth root of the eighth power of minus seven of course that's the 7 that I predicted it would be a moment ago so that's one little kink you have to get used to never take an even power of a negative number and if you do have a power underneath like this the even powers will knock out the negatives they disappear now there's an important special case back to square roots again important special case of that fact that I just mentioned and that case is the following if you take the square root of x squared now that's the same theorem we just mentioned it says this should be the absolute value of x well look what that gives us that gives us a very simple way to write the absolute value of x in terms of powers and roots so this is sometimes handy this you know you can think of this as these box squared square root is the absolute value of box so anything that you can put in there the absolute value can be rewritten this way so this is an alternative notation or an alternate notation for the absolute value of x and as an altered notation sometimes has its uses but let me write one more thing down here so I don't confuse anyone recall that this is not the same thing as taking the square root of x and squaring it that is by definition equal to X so the square root of x quantity squared is X however if you reverse the operations the square root of x squared you get the absolute value of x and this this is handy so it's worth paying attention to all right we're talking about nth roots and rational exponents we haven't talked about the rational exponents parts of that so let me go ahead and discuss that now we have one example in a moment ago when I talked about square roots in a previous bullet I said square roots can be written as X to the one-half power well now nth roots you might expect could be written as X to the 1 over N power we now write the nth root of a as a to the 1 over N power just as before we wrote the square root of a was equal to a to the one half so this is a new notation what does this mean it's always good to keep writing down what this means this means if you take the nth root of a to the nth power that of course is the same now as a to the 1 over N to the nth power but that's a to the N over N and just as before we see that that's just a so it will turn out that using this power this rational exponent notation is going to make things a lot easier let's go back to the list and see what's next we've looked at square roots we've looked at nth roots and rational exponents now finally let's look at operations with rational exponents so operations with rational exponents we did this before with integer exponents and so a lot of this will be similar the only difference is instead of integers we're now dealing with rational numbers but the same kind of rules whole thank goodness and you know what else we can call this we can also say this is operations with rational exponents and translations away from radicals radicals are those root expressions because that is another one of my rules of thumb that I'm going to give you a tip do not in most cases leave problems in radical form write them as exponents that are rational you'll find it to be much easier so here's the first of the rules suppose you have a to the M power and you take the nth root of it so M underneath and througt well what is that that turns out to be the same thing first of all as reversing it the nth root of a to the M power that will be written as a to the M over N because M is the power one over N is the way we write roots and if you want to confirm that that's a to the 1 over N to the M power also can be written as a to the M to the 1 over N power now apart from those special cases we mentioned earlier where you have negative numbers and even roots this is a perfectly good rule and you'll find it useful and we'll try it in a few examples later let me show you a few more of these rules they really will gel when you start doing examples suppose you'd have a and you take the nth root of it and then you take the and through to that well the inside part is a to the 1 over N and then you're taking it to the 1 over m power so the result is you have a to the 1 over m m and that is the way you want to remember this immediately get away from the radical notation and get it into the exponential notation with rationals it's much much nicer let's do a couple more here just to complete the list suppose you take the nth root of a be rewriting that that's a B to the 1 over N power and then using the same rules we use before you can rewrite this as a to the 1 over N times B to the 1 over N and to do the quotient version a over B and of course whenever I write a over B I'm going to assume that the bottom is not 0 so we don't have nonsense this can be written as of course a over B to the 1 over N and then the top is a to the 1 over n over B to the 1 over N now as these rules pass you by at what seems like the speed of light what you want to see is why they seem to be true and then you want to practice them I'm going to now give you some examples that will illustrate some of this but this won't make up for your own efforts a little bit later here's the example again the object here is simplified so we can use some of these rules let's start off with a couple of easy things 8 to the 2/3 power while that's complicated looking let's see if I can simplify it and make it a little easier to work with if I rewrite this as 8 to the 1/3 power squared of course that's the same thing as the original 2/3 this is 2 times 1/3 I noticed something now I didn't notice before I know from my experience that 8 is the cube of 2 so the cube root of 8 is 2 so now I have 2 inside squared well that's easy that's 4 so what looked like a difficult number to begin with 8 to the 2/3 is really just a very familiar number 4 let's do some more like this how about 16 to the three-halves now again I've chosen this so that things will work out in my favor if I rewrite this as 16 to the 1/2 cubed I now recognize that 16 is a perfect square so I can take the square root and get 4 cubed and 4 cubed is another number that's not too unfamiliar 64 all right well those were particularly nice let me do one that is not nice suppose I have an expression like minus 8 times X to the fifth all to the one-third power now let's see if I can simplify this a little bit first of all I'm going to run the 1/3 over each of the factors as the rule says I can do minus 8 to the one-third times X to the fifth to the one-third and I will continue my equal signs down here where can I go from here well really not too much farther I might recognize that minus 8 has a cube root 1 over 3 which is minus 2 so I can write minus 2 here times X to the well there's nothing I can do with this X to the fifth to the one-third of I could write it as X to the 5/3 if I like that's a perfectly good way of stopping some people prefer however that you take it one step further and write minus two and then realize that the X to the 5/3 can be re-written as X to the three thirds times X to the 2/3 remember how this works if you have two bases that are the same you add the powers if you can see my little two here two thirds and three thirds of course add back to 5/3 the advantage to this is you and I have minus 2x to the 3/3 as X and then times X to the two thirds now I only gave you both of those because there's different tastes in the world I prefer frankly the first one we're only a single power of X occurs and it's an improper traction is there as the rational exponent but that doesn't bother me so there's a series of a few examples let's look at some more that involve more variables so that we get some experience here example another one that's simplify and here is the original X to the fifth Y under the square root symbol all over X cubed Y cubed under the square root symbol now I'd like to simplify that my first instinct is to get rid of the square roots but actually it pays to leave them there for just a moment and to realize the square root on the top and the square root of the bottom this can all be written as the square root of the entire fraction X to the fifth Y X cubed Y cubed now we can rewrite that square root as one half power and we can also notice inside that things are going to work out nicely if I have X to the fifth on the top and X cubed on the bottom that leaves me with x squared on the top remember it's five minus the three here I have Y over Y cubed I can take Y here and cancel it or write Y over Y to get one and be left with Y squared on the bottom I can reinterpret the inside as x over Y quantity squared all taken to the one-half power now this is all even powers which means in taking a root like this as the rule said we had a theorem earlier that said what I should end up with is the absolute value of x why because the inside square would remove any negative any negative numbers that appeared in here and then the square root would not recover them so I don't know what's inside the best I can do is write down x over Y absolute value now look at that that's much simpler than the original one that after all is the point all right let me do one more that is more complicated than any of the ones we've seen before I come back and give you a list of tips so this is a longer exercise and this is big so let's write this down square root of 1 plus X minus X times 1 over 2 square root of 1 plus X and all of that is over 1 plus X all right and we will also assume here I'll put it over here we will assume that X is greater than or equal to 0 that way if any absolute values turn up the absolute value of a number that 0 or bigger is just itself and I won't have to worry about that all right here's my first piece of advice this is a big fraction here's a hint rewrite it this way 1 over 1 plus X there's the denominator think of it as a fraction 1 over that times the large part of the top so I'll just copy the top square root of 1 plus X minus and this X here can be written on the top of this fraction so I'm going to do that x over 2 square root of 1 plus X now see I've already made some simplifications that make life easier for me the X here I have moved up to the top that's what appears here and I've rewritten this so that I have 1 over this out front now you see how this works this numerator if I multiplied it back out would multiply times 1 and I'd get the fraction back but now it's out of my way temporarily and I can just concentrate on this well what I need to do here is to add this into a single fraction the problem is the first one has no denominator so the second denominator is the one I'm going to be looking for how will I get that well the best idea is to multiply the first thing here remember we can think of this as square root of 1 plus X over 1 multiply top and bottom here by the same thing so that I don't change the problem the same thing being this number here so what I will do is put into here right there we'll multiply 2 square root of 1 plus x over 2 square root of 1 plus X now that's 1 so that won't change the problem but it will give this the proper denominator so let me continue on to another page I'll bring this one back so you'll see what I'm what I've just done but now I have repeating this 1 over 1 plus X that doesn't change but now inside here all of this has the same denominator 2 square root of 1 plus X that's by design on the left on the right rather I have minus X that's what was there before and on the right I have 2 times the square root of 1 plus x squared now let me bring the previous sheet back so you can follow this the front 1 over 1 plus X has not changed in here I've multiplied top and bottom of the first fraction by these numbers the bottom makes the denominator the same as this so I can add the fractions on the top I'm left with 2 times this times this 2 times square root of 1 plus x squared on the right I have a minus X so remembering that see what I have here 2 square root of 1 plus x squared minus X ok now let's continue notice with this single fraction on the left and the right I can multiply the two denominators and I can multiply the two numerators that's how you multiply fractions so I'm going to get one large fraction on the bottom I'll have 2 and what happens when I multiply 1 plus X times the square root of 1 plus X well think about that for a moment 1 plus X is 1 plus X to the 1 power square root of 1 plus X is 1 plus X to the 1/2 power so when I multiply this by that I end up with 1 plus X to the 1 plus 1/2 power which is 3 halves now what happens on the top here 2 stays square root of 1 plus x squared by its very definition is going to be 1 X now there's a place where I might have needed absolute value if I hadn't chosen my X to be positive to begin with minus X well the top can be simplified this is 2 and then 2 X minus X gives me plus X on the bottom I have 2 times 1 plus x to the three-halves and that's where I am going to stop there's no other cancellation that seems apparent to me so I'll just leave it like that and this is much simpler than let's bring the original back hope that's that is the original the original is this large fraction to begin with and I have now simplified it into this now the more work that you do the more you will appreciate these kinds of operations okay let me go back and now start giving you some more advice and end up with a few tips on working with rational expressions here's a note that I'm going to make because this is something that people get upset about and I think we can make this perfectly simple and I'll do it with an examples you see what I mean I will write down 1 over square root of 2 now that is a perfectly good fraction and it's simple one square root of two not very hard but however some people and I am NOT one of them some people prefer we write the following instead of the nice simple elegant one over square root of two they want you to remove square roots from the denominator multiplying the top and bottom by square root of two we'll do that because the top will be square root of two in the bottom square root of two times itself of course is two they prefer this form on the Left I prefer the simpler form this technique is called rationalizing the denominator rationalizing I can spell it correctly here rationalizing the denominator this is not crucial this is not overly important why was this invented in the first place because calculations like this used to be done by hand and when you do things by hand it does help to have a rational number or just an integer in the denominator however with the advent of calculators and computer algebra systems this number up here which is more elegant and simple is just as easy to work with as this one so although this is a technique that exists I'm not going to stress it and you shouldn't worry too much about it alright well we've had a lot of notation now and before I leave you I want to first do a notation summary just so you are clear on what these notations mean and don't mean the first one of course I'll remind you that X to the 0 equals 1 as long as X is not 0 well since we've raised the question of 0 what about 0 well let me give you all the properties of 0 that we've been covered in this section plus a few others from before and let me take a particular number to do it 5 times 0 as you know is 0 0 over 5 as you know is 0 0 to the fifth I hope as you know is 0 what about 0 to the minus 5 power I'm going to put that in quotes because that's undefined why is it undefined because the minus in the exponent means this is 1 over 0 to the fifth and you can't divide by 0 what is 0 to the 1/5 it is 0 what is 0 to the 0 again this is undefined as we indicated earlier so there are all your basic zero facts so you might say that we've just discussed nothing all right continuing the notation for numbers that are not 0 here's what you can say just to get this all clear in your mind okay 5 times X means what it means X plus X plus X plus X plus X right you add X 5 times x over 5 is the same as 1/5 times X and notice be very clear this is not the same as 1 over 5x again it's a matter of grouping this is 1/5 times X this is not C this is 1/5 times 1 over X so these are definitely not the same thing then there's X to the fifth and what was that that's x times X X times X times X so it's the product of five X's what was X to the minus fifth this is not negation it is not subtraction it says right X to the fifth on the bottom and finally what is X to the one-fifth well that is by definition the fifth root of x that's the number you multiply by itself five times to get X now some tips and then you'll have a chance to practice on your own tips you want to jot these down always another one of my always remarks always rewrite square root or the nth root of a as a to the 1 over N first do that first and then proceed with whatever problem you're working on ok return to this form this radical form only if it is requested of you otherwise leave things in the rational form box to 1 over N is much easier to work with so that's a piece of advice from my experience and then finally remember that one case that we have difficulty with when you have an even root of a negative number okay even root of a negative number is not real in fact it's what's called a complex number and we will talk about that again but for now we've had a lot of information about nth roots and rational exponents and I think it's time for you to take a little time of your own and practice these you

Channel: UMKC

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Keywords: Integer exponents, base, Handy lemma, negative integers, square roots, rational exponents.

Id: SQI97IAUqo8

Length: 92min 36sec (5556 seconds)

Published: Mon May 04 2009