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# Powers of products & quotients (integer exponents)

## Video transcript

do some example raising exponents or products of exponents to various powers especially when you're dealing with integer exponents so let's say we have 3 to the negative 8 times 7 to the third and we want to raise that to the negative 2 power and I want you to pause this video and see if you could simplify this on your own so the key realization here there's a couple of ways that you can tackle it but the key thing to realize is that you have the product of two things and then you're raising that to some type of a exponent that is going to be the same thing as raising each of these things to that exponent and then taking the product so this is going to be the same thing as 3 to the negative 8 and then that to the negative 2 times 7 to the third to the negative 2 so I'll do 7 to the 3rd right over here and if I want to simplify this 3 to the negative 8 to the negative 2 we have the other exponent property that if you're raising to an exponent and then raising that whole thing to another exponent that you can just multiply the exponents so this is going to be 3 to the negative 8 times negative 2 power well negative 8 times negative 2 is positive 16 so this is going to be 3 to the 16th power right over there and then this part right over here 7 to the 3rd to the negative 2 that's going to be 7 to the 3 times negative 2 which is 7 to the negative sixth power so that is 7 to the negative 6 and this would be about as much as you could simplify it you could rewrite it different ways 7 to the negative 6 the same thing as 1 over 7 to the sixth so you could write it like 3 to the 16th use that same shade of blue 3 to the 16th over 7 to the sixth but these two are equivalent and there's other ways that you could have tackled this you could have said that this original thing right over here this is the same thing as 3 to the negative 8 is the same 1 over 3 to the 8th so you could have said this the same thing as 7 to the third over 3 to the 8th and then you're raising that to the negative 2 in which case you would raise this numerator to the negative 2 and the denominator negative 2 but you would have gotten to the exact same place let's do another one of these so let's say let me so let's say that we have we have got a to the negative 2 times 8 to the 7th power we want to raise all of that to the 2nd power well like before I can raise each of these things to the 2nd power so this is the same thing as a to the negative 2 to the second power times this thing to the 2nd power 8 to the seventh to the 2nd power and then here negative 2 times 2 is negative 4 so it's a to the negative 4 times 8 to the 7 times 2 is 14 8 to the 14th power in other videos we go into more depth about why this should hopefully make intuitive sense here you have 8 to the 7th times 8 to the 7th well you would then add the two exponents and you would get 2 8 to the 14 so however many times you have 8 to the seventh Yul's you would just keep adding the exponents or you would multiply by 7 that many times hopefully that didn't sound too confusing but the general idea is if you raise something to exponent and then another exponent you can multiply those exponents let's do one more example where we are dealing with quotients which that first example could have even been perceived as so let's say we have let's say we have 2 to the negative 10 divided by 4 squared and we're going to raise all of that to the seventh power well this is equivalent to 2 to the negative 10 raised to the seventh power over 4 squared raised to the seventh power so if you have the difference of two things and you're raising it to some power that's the same thing as the numerator raised to that power divided by the denominator raised to that power well what's our numerator going to be well we've done this drill before it'd be 2 to the negative 10 times 7th power so this would be equal to 2 to the negative 70th power and then in the denominator 4 to the 2nd power then that raised to the seventh power well 2 times 7 is 14 so that's going to be 4 to the 17th for to the 17th power now we actually could think about simplifying this even more there's multiple ways that you could rewrite this but one thing you could do is any look to before is a power of two so you could rewrite this as this is equal to 2 to the negative 70th power over instead of writing for instead of writing for to the 17th power and why did I write 17th power should be 4 to the 14th power let me correct that instead of writing for to the 14th power I instead could write so this is two you get the colors right this is 2 to the negative 70th over over instead of writing for I could write 2 squared to the 14th power 4 is the same thing as 2 squared and so now I can rewrite this whole thing as 2 to the negative 70th power over well 2 to the 2nd and then that to the 14 plus 2 to the 28th power 2 to the 28th power and so can I simplify this even more well this is going to be equal to 2 to the I can just if I'm taking a quotient with the same base I can subtract the exponent so it's going to be negative 70 it's going to be negative 70 minus 28 power minus 28 and so this is going to simplify 2 to the negative 98 power and that's another way of viewing the same expression