path 100 lesson 10 - deals with rational exponents and before we get into that let's just review the rules that we covered and 10-1 so our rules that were covered in 10-1 are if these squared equals a then B equals the square root of a the square root of a squared is the absolute value of a if B cubed is a then B is the cube root of a the nth root of a to the nth is the absolute value of a when n is even and the nth root of a to the nth is just a when n is odd so these rules have not gone away that we still are going to use those today but this time we are going to have rational exponents I mean you see that little word rational it means fraction Y a fraction exponents so our newest rule for today is going to look like this ok so we've got a raised to the 1 in okay so here's how the rule goes for that this really means the nth root of a to the first power right so my denominator is the root the one up here is that exponent okay that's where that comes from so a to the 1 over N is the nth root of a to the first power excuse me so if I had 64 to the 1/2 that that would mean the square root of 64 to the first power notice I didn't write the 2 or the 1 because when you write square root you don't usually write the 2 and when something's raised to the first you don't usually ride the one alright the square root of eight square root of 64 is 8 let's try another if I have negative 8 raised to the one-third okay this time I am going to write that 3 ok because we do write it when it's not a 2 so we're taking the cube root of negative 8 and again I'm not going to write the 1 because when the exponent is 1 we don't write that all right so the cube root of negative 8 I think about it like this that is negative 2 cubed so now I can use my little rule over here that says this last rule from 10 1 the n32 a to the N is a so I get negative 2 no absolute value is needed because my n value is odd ok now I have all of this 6x squared Y in parenthesis raised to the 1/5 so that 5 is my radical ok that's the root I'm taking and then I'm going to put X 6x squared Y now it's raised to the first power so I don't have to put that one now that cannot be simplified so that is done just like it is all right so I'm just going to give you a couple of problems and what we're going to do is we're going to rewrite each of them using rational exponents so we're going to go backwards okay we're going to go the other direction all right so I've given you the radical and we're gonna rewrite it with a rational exponent so all the stuff that's inside we're gonna put in parentheses and we know there's not a one there because we don't show the one but there is a one so this would be raised to the 1/5 so that would be our answer cube 1380 to the 1/5 you must have the parentheses let's try No okay now let's don't make this difficult take the stuff that's inside all right here's all the stuff that's inside put in parentheses and we raise it to the 1/7 okay that's 7th root now that top number is not always a 1 okay so let me give you another little rule for when that little top number is not a 1 in is still going to be our route and illness still going to be our exponent it's just not always going to be one but another way to write that that is sometimes super helpful is for us to say the square root of a and raise the whole thing to the end now either of those is correct but sometimes it's easier to simplify the second one alright so if I have 1,000 to the 2/3 all right so the 3 is my radical the 2 is my exponent now I can tell you this if you square a thousand there's a good chance you're going to mess up I'll write down zeroes ok so I'm going to do this one differently because we don't have a cube root button on our calculator that we're supposed to be using so I'm going to rewrite this one the second way and put the squared on the outside now the reason I want to do that I can do the cube root of a thousand all right the cube root of a thousand a thousand is 10 cubed so the cube root of 10 cubed is 10 and now it's still squared which gives me 100 not super I know that some of you are like oh just put it my calculator a lot of these are gonna have variables and you're not going to be able to put the variables in your calculator so make sure you're able to do this without having to hit those buttons on your calculator now remember you're supposed to be using a four function calculator that should not have a cube root or a square if it will have a square root it should not have a cube root it certainly shouldn't have an exponent button okay if I take this one we're doing the square root of 16 Q again I can't do 16 cubed now on your little calculator you could say 16 times 16 times 16 but then I can't take the square root of that so I'm not going to do it that way I'm going to write the square root of 16 and I'm going to put the cubed on the outside because I can do the square root of 16 it is 4 now I can cube for 4 times 4 times 4 is 64 so that would be my answer okay now big mistake here that people make is they go oh that's negative 32 raised to the 3/5 and it is not let me write it differently these are two different problems this one says negative 1 times 32 to the 3/5 so this would be negative fifth root of 32 cubed this one would be the 5th root of negative 32 cubed big difference in the way that that is initially written all right so let's do both of them right here this negatives on the outside the fifth root of two to the fifth all right so the fifth root of 2 to the fifth is 2 cubed now the 2 is the only thing cubed I get negative 8 all right so this is where some of you're gonna be like why did you even bother stick with me when the rules is what's important here all right so the fifth root of negative two to the fifth is still negative 2 and that is cubed which is negative 8 now I know that we got the same answer on both of these however is that had been an even read if the three had been a four okay now we've got a big difference if the three had been a two then one of these answers would be positive and one of them would be negative so you have to be careful those were two different problems all right so we're going to go backwards again and we're going to rewrite using rational exponents this time I give you the radical and you change the way that it looks okay so I'm giving you the radical now we're gonna ride it with rational exponents so this would be 6 to the 4/3 all right let's do another one if I had the fifth root of 2xy all of that raised to the seventh so I'm going to take the 2xy it's gonna be put in parentheses the fifth root that's my denominator and this 7 goes up there in the numerator okay all right so our next little rule says what if our exponent is negative what if our exponent is negative well we've already covered negative exponents and that means put it in the denominator okay now we have learned that a rational exponent means this is the square root of a to the I can raise the whole thing to the N remember I can write it either way okay so let's simplify using that rule if I have 100 to the negative 1/2 that means the negative 1/2 means that the whole thing in the denominator and now it's a positive exponent okay not finished because 100 to the 1/2 means the square root of 100 which is 10 so we get 1 over 10 for our answer these rational exponents tend to get a little tricky for people a negative exponent all it does is puts it in the denominator and now the negative exponent is a positive all right now this says the cube root of 8 to the first power the cube root of 8 is 2 so our answer here is 1/2 okay still simplifying all right now that negative exponent just means put put it in the denominator in the exponent becomes positive all right now we're going to use our radical sons to help us out so we've got the ninth root of three X Y and I can put this five on the outside okay now we get my answer let's see one with just numbers again all right now 32 to the negative three-fifths meet that negative exponent means put it in the denominator and it will become a positive exponent now we take the fifth root of 32 when we take that answer and cube it well the fifth root of 32 is 2 and then we cube that so 2 cubed is 8 so we end up with 1 over 8 all right these just take some practice there are a couple more rules for us okay so let me just remind you of some old rules these are just things we've already done this semester that are going to be helpful if I am multiplying two expressions that have exponents to multiply those I add the exponents if I am dividing two expressions then I would subtract top exponent minus the bottom if I have a power raised to a power I multiply the exponents these are old rules if I have a product raised to an exponent then I raise everything in the parenthesis to that exponent okay even if that was a division the numerator gets raised to the exponent and the denominator gets raised now those are old rules that's nothing new but I just want to remind you before we do our fraction exponents because these rules right here are going to apply so we're still simplifying 6 to the 1/7 times 6 to the 4/7 okay that's this first rule up here okay old rule just an old rule we're multiplying our bases are the same we add the exponents so we add one seven two plus four 7s now we actually do that and we get 5/7 now in MyMathLab you need to pay attention to the directions if it wants you to write your answer as a rational exponent then this would be correct if it wants you to use as a radical expression then that's where we would write it using a radical so pay attention to the directions people tend to miss these just because they don't read the direction as well okay so 50 X to the one-third divided by 10 and X to the 4/3 okay well we would do 50 divided by 10 and we get five now when we're dividing we use the second rule of reminder up there we subtract the exponents top exponent minus bottom exponent so this gives me 5x to the negative 3 over 3 well I can simplify 3 over 3 that's negative 1 okay now that negative wine means put it in the denominator but negative 1 is only on the X so the X goes to the denominator but the 5 does not okay fine okay so I've got 9.1 race to the 2/5 and then the whole thing is raised to the 3/4 so we have a power raised to a power so really this is 9.1 to the 2/5 times 3/4 power raised to a power we multiply the exponents okay now when I'm multiplying these I can simplify 2 goes into itself once it goes into 4 twice so I have 9.1 and now my new exponent is 3/10 so this will be my answer 9.1 to the 3/10 okay so I have X to the negative three-fifths Y to the 1/4 multiplied together and then all of that raised to the 1/3 so a couple ways I can do this but what I'm gonna do is I'm gonna say okay I have a power raised to a power so everything in there gets raised to that power so negative three-fifths gets multiplied by one third power raised to a power you multiply Y to the 1/4 get 2 multiplied by one third power raised to a power gets multiplied now I have to do both of those so I get X to the now when I multiply here I can simplify so I end up with negative one-fifth y to the 112 now that negative one-fifth you can never leave a negative exponent in your answer so that means that that X to the one-fifth has to go to the denominator but the Y to the 112 stays in the numerator so this is what my answer would look like okay all right I've got a couple more and we'll be done all right so my directions are still simplify and I'm gonna add to my directions this time and say these rational exponents this is we're reading the directions is going to be super important in MyMathLab right so I've got the sixth root of x cubed okay so if I rewrite that using rational exponents it would be X to the three six now that can be simplified you can simplify that fraction to say one half okay now this would be rational exponents all right that would be that answer if the directions asked for the answer in radical form it would be the square root of x you would take X to the one-half and you write it as a radical okay so pay attention to do the directions asked for rational exponents or is it asking for a radical form they're different I am asking for rational Xfinity okay so I've got this radical I'm going to write it as rational exponents all right everything inside gets raised to the one-third okay not finished because I need to simplify now everything in parentheses gets raised to the 1/3 power raised to a power I multiply so the 8 to the one-third is to the cube root of 8 is 2 and if I think about this I can simplify that fraction and I get a to the fourth so my solution here would be 2 a to the fourth restaurant some old ones and that is all for section two please make sure you practice this before you try to move on to the assignment for section three