Hello, my name is Anthony Varela. And today we're going to talk about properties of exponents. So we're going to demonstrate a couple of properties of exponents, specifically those involving products, or multiplication, quotients, or division, and then powers themselves, or more exponents. And so the whole utility for these properties of exponents is that we can simplify exponential expressions using these properties.
So, first, let's talk about properties involving products. So I have this expression here-- 4 raised to the third power multiplied by 4 raised to the fifth power. So I'm going to expand this expression. I know that 4 to the third means that I'm using that base number 4 in a chain of multiplication three times. So it's 4 times 4 times 4.
And 4 to the fifth is using that base number 4 in a change of multiplication five times. So I've written that out. 4 times 4 times 4 times 4 times 4. Now notice, because my base is the same, both 4 to the third power and 4 to the fifth power, I have a chain of multiplication that involves the same number, just more times, right? So I'm counting how many times I use the number 4. And I see 1, 2, 3, 4, 5, 6, 7, 8. So I could write this as 4 to the eighth power.
Now what's the relationship between our exponents of 3, 5, and 8? Well, I noticed that I just added 3 and 5 and got to 8. So this is my product of powers property. And this says that a number raised to the power of n times a number of the same base raised to the power of m can be expressed as that base raised to the power of n plus m, so the sum of the two individual exponents.
So that's one of our important properties. Let's write that down in our notes-- the product of powers property. Let's take a look at another property that involves products. I have 18 raised to the power of three. Well, I know that I can write this as 18 times 18 times 18, just using that base number three times in my chain of multiplication.
But what if I wanted to break down 18 into other factors? So I could say, since 18 equals 6 times 3, I could say that 18 cubed is 6 times 3 times 6 times 3 times 6 times 3. Well, then I noticed that I have 6 used as a factor three times and 3 used as a factor three times. So I can write this as 6 cubed times 3 cubed.
And so we call this then the "power of a product property," where if I have a product here, a times b, and I'm raising all of that to a power, I can say that this equals one factor to that power times another factor to this power. And we should note that this works no matter how you break down your number.
I could have broken it down 9 times 2, or 9 times 3 times 3. So this can be expanded. So we're going to write that down then, the power of a product. So now let's talk about properties involving quotients. So here I have 7 raised to the power of 8 and I'm dividing that by 7 raised to the power of 5. So, once again, I'm going to expand our exponential expressions here, both in the numerator and denominator.
7 raised to the power of 8 can be rewritten as using 7 eight times in a chain of multiplication. And same deal with 7 to the fifth. I'm going to write 7 five times in a chain of multiplication.
And now we notice we have common factors in both our numerator and denominator, quite a few of them. And they cancel out because multiplication and division are inverse operations. So really what I'm left with is just three factors of 7, which I can write a 7 to the third power.
You can imagine these first five being canceled. These five being canceled. So we're left with just 1, 2, 3 factors of 7. So what's the relationship then between the 8, the 5, and the 3?
Well, this is very similar to our properties involving products. I can rewrite this but subtract the exponents. So we call this our "quotient of powers property." And notice that this only holds true if our base numbers are the same because they need to be able to cancel out when you expand them.
So we say that a to the n divided by a to the m equals a raised to the difference of n and m. So that's an important property that we're going to put into our notes, the quotient of powers.
Well, there's another property that involves quotients here. I have 3/4. It's commonly referred to as a "fraction," but it's technically a quotient, 3/4. And what if we were to square that whole quantity?
Well, I know that, because we're squaring this, I can write 3/4 times 3/4. That's just using 3/4 as a factor twice in this change of multiplication. And multiplying fractions, I know that we can just multiply across the numerator, multiply across the denominator. So this is also a 3 times 3 divided by 4 times 4, which I can write now as 3 squared over 4 squared.
So notice that we were sort of distributing that power of 2 both into the numerator and the denominator. And this is what we call the "power of a quotient property," where we have some quotient, a over b, and we're raising all of that to a power. We could say that this is a to the n over b to the n. So we're going to write this down as an important property in our notes, the power of a quotient.
Well, there's one more property that I'd like to talk about today. And this involves powers. So we're going to raise something to an exponent and then raise that entire quantity to an exponent again. So here we have 3 squared and are raising that to the power of 3. So I know because we're raising 3 squared to the power of 3, I can use 3 squared as a factor in a chain of multiplication three times. So this is 3 squared times 3 squared times 3 squared.
Well, then I know I can write 3 squared as 3 times 3. So I'm going to be writing that this 3 times 3 corresponds to this 3 squared. This pair of 3's correspond to my middle. And the last pair of 3's corresponds to this 3 squared.
Well, notice I'm just using 3 six times in this chain of multiplication. So I can write this is 3 to the sixth. So now what's the relationship between 2, 3, and 6? We're not adding the powers here. We're multiplying the powers. So this is our "power of a power property."
And this says if we take a number and we raise it to the power of n, and we raise that whole quantity to the power of m, this is the same as saying a raised to the n times m power. So we're multiplying those two powers. So we're going to write that down in our notes as another one of our important properties.
All right. Let's review our notes for today. We talked about several properties of exponents. And it should be noted that these properties also hold true if the exponents are fractions or negative numbers. And these properties were the product of powers property, the power of a product, the quotient of powers, the power of a quotient, and, lastly, the power of a power property.
Well, thanks for watching this video on the properties of exponents. I hope to catch you next time.