Powers of products & quotients (integer exponents)

- [Instructor] Do some example, raising exponents or products of exponents to various powers, especially when we're dealing with integer exponents. So let's say we have three to the negative eight times seven to the third, and we wanna raise that to the negative two 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 couple of ways that you can tackle it, but the key thing to realize is if 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 three to the negative eight, and then that to the negative two times seven to the third to the negative two, so I'll do seven to the third right over here. And if I wanna simplify this, three to the negative eight to the negative two, we have the other exponent property that if you're raising to an exponent and then raising that whole thing to another exponent, then you can just multiply the exponents. So this is going to be three to the negative eight times negative two power. Well, negative eight times negative two is positive 16, so this is gonna be three to the 16th power right over there, and then this part right over here, seven to the third to the negative two. that's gonna be seven to the three times negative two, which is seven to the negative sixth power. So that is seven to the negative six, and this would be about as much as you could simplify. You could rewrite it different ways. Seven to the negative six is the same thing as one over seven to the sixth, so you could write it like three to the 16th. We'll use that same shade of blue, three to the 16th over seven 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, three to the negative eight is the same thing as one over three to the eighth, so you could have said this is the same thing as seven to the third over three to the eighth, and then you're raising that to the negative two, in which case you'd raise this numerator to the negative two and the denominator to negative two, but you would have gotten to the exact same place. Let's do another one of these. So let's say that we have got A to the negative two times eight to the seventh power, and we wanna raise all of that to the second power. Well, like before, I can raise each of these things to the second power, so this is the same thing as A to the negative two to the second power times this thing to the second power. Eight to the seventh to the second power, and then here, negative two times two is negative four, so that's A to the negative four times, eight to the seven times two is 14, eight to the 14th power. In other videos, we go into more depth about why this should hopefully make intuitive sense. Here you have eight to the seventh times eight to the seventh. Well, you would then add the two exponents, and you would get to eight to the 14th, so however many times you have eight to the seventh, you would just keep adding the exponents, or you would multiply by seven 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 two to the negative 10 divided by four squared, and we're gonna raise all of that to the seventh power. Well, this is equivalent to two to the negative 10 raised to the seventh power over four 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 a 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 two to the negative 10 times seventh power, so this would be equal to two to the negative 70th power, and then in the denominator, four to the second power, then that raised to the seventh power. Well, two times seven is 14, so that's going to be four 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 say, "Hey, look, "four is a power of two." So you could rewrite this as this is equal to two to the negative 70th power over, instead of writing four to the 17th power, why did I write the 17th power? It should be four to the 14th power. Let me correct that. Instead of writing four to the 14th power, I instead could write, so this is two, get the colors right. This is two to the negative 70th over, instead of writing four, I could write two squared to the 14th power. Four is the same thing as two squared, and so now I can rewrite this whole thing as two to the negative 70th power over, well, two to the second, and then that to the 14th, well, that's two to the 28th power, two to the 28th power. And so can I simplify this even more? Well, this is going to be equal to two to the, if I'm taking a quotient with the same base, I can subtract the exponent. So it's gonna be negative 70. It's going to be negative 70 minus 28th power, minus 28, and so this is going to simply two to the negative 98th power, and that's another way of viewing the same expression.