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# Exponent properties with quotients

CCSS.Math:

## Video transcript

let's do some exponent examples that involve division let's say I were to ask you what five to the sixth power divided by 5 to the 2nd power is well we could just go to the basic definition of what an exponent represents and say well five to the sixth power that's going to be 5 times 5 times 5 times 5 times 5 one more five times five five times itself six times when five squared that's just five times itself two times so it's going to be five times five well we know how to simplify a fraction or a rational expression like this we can divide the numerator and the denominator by one five and then these will cancel out and we can do it by another five or this five and this 5 will cancel out and what are we going to be left with 5 times 5 times 5 times 5 over well you could say over 1 or you could say that this is just 5 to the fourth power now notice what happens essentially we started with 6 in the numerator 6 5s multiplied by themselves in the numerator and then we subtracted out we were able to cancel out the 2 in the denominator so this really was equal to 5 to the sixth power minus 2 so we were able to subtract the exponent the denominator from the exponent in the numerator and let's remember how this relates to multiplication if I had 5 to the let me do this in different colors five to the sixth times times 5 to the second power we saw in the last video that this is equal to 5 to the 6 plus I'm trying to make it color-coded for you 6 plus 2 power now we see a new property and in the next video we're going to see that these aren't really different properties that they're kind of same sides of the same coin when we learn about negative exponents but now in this video we just saw that 5 to the 5 to the 6th power divided by 5 to the second power five to the let me do it in different color 5 to the second power is going to be equal to 5 to the it's not it's time-consuming to make it color-coded for you 6 minus 2 power 6 minus 2 power or 6 or 5 to the fourth power here it's going to be 5 to the 8th so when you multiply exponents with the same base you add the exponents when you divide with the same base you subtract the numerator exponent you subtract the denominator exponent from the numerator exponent let's do a bunch more of these examples right here let's do a bunch more what is 6 to the seventh power divided by 6 to the third power well once again we can just use this property this is going to be 6 to the 7 minus 3 power which is equal to 6 to the 4th power and you could multiply it out this way like we did in the first problem and verify that it indeed will be 6 to the 4th power now let's try something interesting let's try something interesting and this will be a good Segway into the next video let's say we have 3 to the 4th power divided by 3 to the 10th power well if we just go from basic principles this would be 3 times 3 times 3 times 3 all of that over 3 times 3 we're gonna have 10 of these 3 times 3 times 3 times 3 times 3 times 3 how many's that 1 2 3 4 5 6 7 8 9 10 well if we do what we did in the last video this 3 cancels with that 3 those threes cancel those threes cancel those 3 cancel and we're left with 1 over 1 2 3 4 5 6 threes so 1 over 3 to the 6th power right we have 1 over all of these threes down here but that property that I just told you would have told you that look this should also be equal to 3 to the 4 minus 10 power well what's 4 minus 10 we get a negative number this is three to the negative sixth power so using the property we just saw you would get three to the negative six power just multiplying them out you get one over three to the sixth power and the fun part about all of this is these are the same quantity so now you're learning a little bit about what it means to take a negative exponent 3 to the negative six power is equal to 1 over 3 to the sixth power and I'm going to do many many more examples of this in the next video but if you take anything to the negative power so a to the negative B power is equal to one over a to the B that's one thing that we just established just now and earlier in this video we saw that if I have a to the B over a to the C that this is equal to a to the B minus C that's the other property we've been using now what we've just learned and what we learned in the last video let's do some more more complicated problems let's do some more complicated problems let's say I have a to the third B to the fourth power over a squared B and all of that to the third power well we can use the property we just learned to simplify the inside this is going to be equal to a to the third divided by a squared that's a to the 3-2 power right so this would simplify to just an A and you can imagine this is a times a times a divided by a times a you'll just have an A on top and then the B B to the fourth divided by B well that's just going to be B to the third right this is B to the first power 4 minus 1 is 3 and then all of that in parenthesis to the third power I don't want to forget about this third power out here this third power is this one let me color code it that third power is that one right there and then this a and orange is that a right there and I think we understand what maps to what and now we can use the property that when you multiply something and take to the third power that's equivalent of taking each of the this is equal to a to the third power times B to the third to the third power and then this is going to be equal to this is equal to a to the third power I'll do it right this a to the third power that's just a to the third right there and then what is this times B to the 3 times 3 power times B to the 9th and we would have simplified this about as far as as you can go let's do one more of these because I think they're good practice and super valuable experience I think later on let's say I have 25 X Y to the sixth over over 20 20 y to the fifth Y to the fifth Y to the fifth x squared so once again we can rearrange the numerators and the denominators so this you could rewrite as 25 25 over 20 times x over x squared right we could have we could have made this bottom 20 x squared Y to the fifth it doesn't matter the order we do it in times y to the sixth over y to the fifth and let's use our newly learned exponent properties and actually just simplifying fractions 25 over 20 if you divide them both by 5 you're going to get this is equal to 5 over 4 X divided by x squared well there's two ways you could think about it that you could view as X to the negative one but you know you have a first power here 1 minus 2 is negative 1 so this right here this right here is equal to X to the negative 1 power or it could also be equal to 1 over X these are equivalent so let's say that this is equivalent to 1 over X just like that and it would be x over x times X one of those sets of X's would cancel out and you're just left with 1 over X and then finally Y to the sixth over Y to the fifth that's why that right there is to the 6-5 power which is just Y to the first power or just Y so times y so if we want to write it all out is just one combined rational expression you have five times one times y which would be five y all of that over 4 times X right there's this is y over 1 so 4 times X times 1 all of that or 4 X and we have successfully simplified it