luster of math unit three lesson three is called exponents that are unit fractions so when it says unit fractions um really important to understand that unit fractions means one in the in the numerator okay so exponents that are unit fractions so 1 12 13 14 Etc okay so exponents if I had like you know know everybody here knows 10 to the 1 is 10 really even 10 to the zero actually 10 the zero is one 10 squar is 100 I'm sorry 10 10 to the 1 is 10 sorry 10 to the 2 is 100 10 cubed is a th okay and so forth but the problem is what happens if I have like in between here what if I have like 10 to the one2 what's that equal to so that's what we're going to learn about today so our goal I can write square and cube roots as exponents and I understand understand that b to the one/ N is equal to the nth root of B so I'm kind of rooing the surprise for you today but um that is part of our goal to make sure we understand that all right so find the solution to each equation so gonna square root both sides now some of you might say five but it's actually five and negative five anytime you have to insert a square root into the equation you need a plus and minus so here so instead of I'll just shortcut I'll write as plus and minus Square < TK of 7 now you can leave it as Square Ro of s because that's exact if you wanted to get give an estimate we could use a little squiggly equal sign that means approximately I get 2.65 all right and I need the plus and minus as well so that means I have two answers now if it's cubed we cube root both sides so hopefully you know where the cube root button is on your calculator um obviously everybody has different calculators so I can't really tell you where that is on your calculator um and show you but uh the answer there is going to be two now if it's an odd root you don't need the plus or minus right and why is that why do you need the plus or minus I guess if you go back to this first one x^2 equals 25 if I plug five in there 5^2 = 25 and then ne5 you gotta technically you got to plug these both in with parentheses so this is5 * and everybody should know a negative times a negative is a positive and it doesn't really work with the cube so like if I have y Cub = 8 if I if I put positive2 and negative -2 -2 cubed is -2 * -2 * -2 which is is equal to8 which is not equal to 8 okay all right so moving on to the next one okay the next one is another Cube so I cube root and this is going to be W equal the cubed root of 19 and as a decimal it's going to be 2.67 okay not a plus or minus because it's an odd odd root all right so we're gonna kind of play around with this to the half power like I did on the very first slide Claire said I know that 9 SAR is 9 time 9 and nine to the first is Nine and Nine of Zer is one I wonder what nine to the 1 half means so here's what she did she graphed y = 9 to the X for some whole number values of X so if you you could graph it on decimos or on your your graphing calculator um you know X and Y if I were to make a table here though just take a look here we know when X is zero we said it was one when X is one we said nine um when X is two it's 81 right so obviously one half has got to be somewhere in between one and nine okay maybe on a graph we could kind of connect the dots and see what happen so here I I graphed it for you whoops so here's yals 9 x so I need to go to 12 so I'm gonna take a look here on my graph and I'm going to see that2 appears to be three right there there's my point right there at three so for a graph function yourself what do you estimate I'm gonna say three so using the properties of exponents Claire evaluated 9 to the 12 time 9 to2 what did she get so I'm not using the graph I'm not using anything other than exponent property properties so if you remember if you have like x to the m * X the N it's x to the M plus n you add the powers so as long as your bases match up you can add the powers so 9 to the 12 * 9 12 is 9 to the 12+ 1 12 is 1 okay so this would be 9 to the 1st so we're just call it nine so for that to be true what values of U what must the value of 9 to the one 12 be so basically we have if you look here it's kind of like we have something times something equals 9 so if I said x x x we'll just pretend this is x equals 9 it's really kind of like x^2 = 9 so your answer would beot both sides I get x equals positive andg -3 but when I look at my graph I see it's only positive there's no negative3 um here so I'm G to just say it's got to be three which is kind of like the square root of nine which is what I got right here awesome let's try it for something oh let's try for a different number here okay so this is yals 3 to the X okay so graph the function yourself what do you estimate to get for 3 to the 12 so X is 1 12 let's take a look at our graph here [Music] 12 is right around here which I can get my line a little straighter there we go looks like H I don't know 1 point I don't know 1.67 let's just estimate it here so I'm going to put a little estimate sign there next use exponent rules to find this so we said 3 12 raised to the second what did he find well I would multiply these Powers right remember um x to the m to the n = x the M time n you multiply so that's going to equal three then he said that looks like a root what do you think he means I think he means that if I wanted to find x to the 12 you could square root both sides to get rid of the square and we could say oh okay three to the one2 because my square root gets rid of this squared equals square root of three so that makes me think oh okay 3 to the one 12 is square root of three and I think it would work for any value okay so so um here's what I'm going to put here 3 to 12 equals squ < TK of 3 I think it would work for anything let's try another number 6 to the2 well if I square that it's going to equal 6 to the 1st which is six and then if I wanted to get rid of the square I'd have to square root both sides which would make this be 6 to the 12 equals the square OT of six so that's showing me that any one half power is equal to a square root okay so you know just make sure you understand here if I have like a to the 12 is equal to the square root of a any value all right so here's it actually works for cube roots fourth root fifth root and so forth so write the pause the video write this down B to the 1 over n is equal to the nth root of B okay and as I keep going here I guess maybe I should have used b instead of a to kind of stay consistent with that let's change that b to the2 and then it's kind of going to be the same thing for a one3 power it's going to be Cub root and oh I didn't write the two here when it's a square root we don't put the two there but we know it's a square root it's there um B to the one fourth power is the fourth root of B sorry I missed that all right and so forth all the way down until you get to B to the 1 over n nth root of B okay all right use exponent rules and your understanding of roots to find the exact value so this is going to be the square root of 25 which is a perfect square so I get five um this is going to be the square root of 15 now I don't know what that is off top of my head if I wanted to use a calculator well how about this before using a calculator let's at least estimate it so um I know that 15 to the zero is one 15 to the 1 is 15 so 15 to the 1/2 it's somewhere between 1 and 15 um let's go ahead type in my calculator square otk of 15 is 3 87 so actually I'm gonna inste of equals I'm going to make an approximate so this one is a approximate this is exact you can leave it as the square root of 15 so this is going to the Cub root of 8 which is two and this is going to be the Cub root of two which this is exact um if I wanted to take my calculator I could type in two to the 1/3 or cubed root of two it'll be the same value um 1 26 all right match each expression to an equivalent expression Okay so let's start with the whole numbers this is easy this is one this is seven this is 49 this is 7 * 7 * 7 343 all right so this is going to be a negative that flips it so it's 1 over 7 do you remember um if I have B to the Nega n that's equal to 1 over B to the N so if I had 7 to the negative 1 it's equal to 1 over 7 the 1st which is 17th this is 1 over 7^ squared right 7 the -2 1 over 7 the positive2 which is 1 over 49 17 sorry 1 over 49 -3 that' be 1 over 343 7 to the 1/2 that's going to be the square root of seven so there it is s to the negative2 okay so 7 to thega - one2 is 1 over 7 to the positive2 so remember 1/2 is square root so it's one over theun 7 which would be this one right here me erase that 1/3 would be cubed root of seven and then this would be one over the cubot of 7 okay there we go how could you explain the meaning of the 1/2 exponent in 3 to the 1/2 to a student who is absent for the lesson so they could understand the idea fully I would basically say all right if I have three to the one2 and I want to know what that equals um if I Square it I now get let's leave that here this is I'm gonna leave it like this but I can actually say what it's equal because it's if you multiply them it's 3 to the 1st okay and I don't really need the one there but and then if I want to undo the squared I could square root both sides so I get back to 3 to the2 equals the square root of three so I would say a 1 half power is equal to the square root all right what would be another way to write 13 to the 1 so since it's 1 it's going to be fifth root of 13 all right so I can write square and cube roots as exponents and really any number that has a um unit of one in the numerator that's a fraction all right write the solution to the x^2 = 5 two ways one using exponents one using radicals so square root square root so since I had to put the square root in I need a plus or minus if the square Root's already there you don't need a plus or minus so I got plus or minus radical 5 plus and minus and then um using exponents I'm going to say plus or minus 5 to the 12 and this is just going to be 78 to the 1/3 okay all right I hope you enjoyed this video thank you for watching