Exponent properties 3

Simplify x to the third, and then that raised to the fourth power times x squared, and then that raised to the fifth power. Now, here we're going to use the power property of exponents, sometimes called the "power rule." And that just tells us if I have x to the a, and then I raise that to b-th power, this is the same thing as x to the a times b power. And to see why that works, let's try that with a-- well, I won't do it with these right here. I'll do it with a simpler example. Let's say we are taking x squared and then raising that to the third power. Well, in that situation, that literally means multiplying x squared by itself three times. So that literally means, x squared times x squared times x squared. I'm taking x squared and I'm raising it to the third power. Now, what does this mean over here? Well, there's a couple of ways to do it. You could just say, look, I have the same base, I'm taking the product, I can just add the exponents. So this is going to be equal to-- let me do it in that same magenta-- this is equal to x to the 2 plus 2 plus 2 power, or essentially x to the 3 times 2. This right here is just 3 times 2. So we get x to the sixth power. And if you want, you say, hey, Sal, I don't understand why you can add those exponents. You just have to remember that x squared-- this thing right over here, we could rewrite as x times x times x times x-- in parentheses, I'm putting each of these x squareds-- times x times x. And this is just x times itself 6 times x times x times x times x times x times x. This is just x to the sixth power. So that's why we can add the exponents like that. So let's just use that power property on this expression right over here. We start off with we have x to the third raised to the fourth power. So that's just going to be x to the 3 times 4 power, or x to the 12th power. Then, we're multiplying that by x squared raised to the fifth power. Well, that's just going to be x to the 2 times 5 power, or x to the 10th power. And now we have the same base, and we're taking the product, we can just add the exponents. This is going to be equal to-- this whole expression is going to be equal to x to the 12 plus 10th power, or x to the 22nd power. And we are done.