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# Multiplying & dividing powers (integer exponents)

CCSS.Math:

## Video transcript

let's get some practice with our exponent properties especially when we have integer exponents so let's think about what 4 to the negative 3 times 4 to the fifth power is going to be equal to and I encourage you to pause the video and think about it on your own well there's a couple of ways to do this one you say look I'm multiplying two things that have the same base so this is going to be that base 4 and then I add the exponents 4 to the negative 3 plus 5 power which is equal to 4 to the second power and that's just a straightforward exponent property but you can also think about why does that actually make sense 4 to the negative 3 power that is 1 over 4 to the 3rd power or you could view that as 1 over 4 times 4 times 4 and then 4 to the 5th that's 5 4 is being multiplied together so it's times 4 times 4 times 4 times 4 times 4 and so notice when you multiply this out you're gonna have five fours in the numerator and three fours in the denominator and so three of these in the denominator going to cancel out with three of these in the numerator and so you're going to be left with five minus three or negative three plus five fours so this 4 times four is the same thing as 4 squared now let's do one with variables so let's say that you have a to the negative 4 power times a to the let's say a squared what is that going to be well once again you have the same base in this case it's a and so and since I'm multiplying them you can just add the exponents so it's going to be a to the negative 4 plus 2 power which is equal to a to the negative 2 power and once again it should make sense this right over here that is 1 over a times a times a times a and then this is times a times a so that cancels with that that cancels with that and you're still left with one over a times a which is the same thing as a to the negative two power now let's do it with some quotients so what if I were to ask you what is what is 12 to the negative 7 divided by 12 to the negative 5 power well when you're dividing you subtract exponents if you have the same base so this is going to be equal to 12 to the negative 7 minus negative 5 power you're subtracting the bottom exponent and so this is going to be equal to 12 to the well that a negative subtracting a negative is the same thing as adding the positive a 12 to the negative 2 power and once again we just have to think about why does this actually make sense well you can actually rewrite this 12 to the negative 7 divided by 12 to the negative 5 that's the same thing as 12 to the negative 7 times 12 to the fifth power if we take the reciprocal of if we take the reciprocal of this right over here you would make the exponent positive and then you get exactly what we were doing in those previous examples with products and so let's just do one more with variables for good measure let's say I have X to the negative 20 power divided by X to the fifth power well once again we have the same base and we're taking a quotient so this is going to be X to the negative 20 minus 5 because we have this one right over here in the denominator so this is going to be equal to X to the negative 25th power and once again you could view our original expression as X to the negative 20th and having an X to the fifth in the denominator dividing by X to the fifth is the same thing as multiplying by X to the negative 5 and so here you just add the exponents and once again you would get X to the negative 25th power