# Exponent properties 1

## Video transcript

Simplify 3a times a to the fifth times a squared. So the exponent property we can use here is if we have the same base, in this case, it's a. If we have it raised to the x power, we're multiplying it by a to the y power, then this is just going to be equal to a to the x plus y power. And we'll think about why that works in a second. So let's just apply it here. Let's start with the a to the fifth times a squared. So if we just apply this property over here, this will result in a to the fifth plus two-th power. So that's what those guys reduce to, or simplify to. And of course, we still have the 3a out front. Now what I want to do is take a little bit of an aside and realize why this worked. Let's think about what a to the fifth times a squared means. A to the fifth literally means a times a times a times a times a. Now, a squared literally means a times a. And we're multiplying the two times each other. So we're multiplying these five a's times these two a's. And what have we just done? We're multiplying a times itself five times, and then another two times. We are multiplying a times itself. So let me make it clear. This over here is a to the fifth. This over here is a squared. When you multiply the two, you're multiplying a by itself itself seven times. 5 plus 2. So this is a to the seventh power. a to the 5 plus 2 power. So this simplifies to 3a times a to the seventh power. Now you might say, how do I apply the property over here? What is the exponent on the a? And remember, if I just have an a over here, this is equivalent to a to the first power. So I can rewrite 3a is 3 times a to the first power. And now it maybe makes it a little bit clearer. A to the first power-- and the association property of multiplication, I can do the multiplication of the a's before I worry about the 3's. So I can multiply these two guys first. So a to the first times a to the seventh-- I just have to add the exponents because I have the same base and I'm taking the product-- that's going to be a to the eighth power. And I still have this 3 out front. So 3a times a to the fifth times a squared simplifies to 3a to the eighth power.