Rewriting exponential expressions as A⋅Bᵗ

- [Voiceover] What I hope to do in this video is get some practice simplifying some fairly hairy exponential expressions. So let's get started. Let's say that I have the expression 10 times nine to the t over two plus two power, times five to the three t. And what I wanna do is simplify this as much as possible, and preferably get it in the form of A times B to the t. And like always, I encourage you to pause this video and see if you can do this on your own using exponent properties, your knowledge, your deep knowledge of exponent properties. All right, so let's work through this together, and it's really just about breaking the pieces up. So 10, I'll just leave that as 10 for now, there doesn't seem to be much to do there. But there's all sorts of interesting things going on here. So nine to the t over two, plus two, so this right over here, I could break this up, using the fact that, and I'll just write the properties over here. If I have nine to the a plus b power, this is the same thing as nine to the a, times nine to the b power. And over here I have nine to the t over two, plus two, so I could rewrite this as nine to the t over two power, times nine squared. All right, now let's move over to five to the three t. Well, if I have a to the bc, so you could view this as five to the three times t, this is the same thing as a to the b, and then that to the c power. So I could write this as, this is going to be the same thing as five to the third, and then that to the t power. And the whole reason I did that is well this is just going to be a number, then I'm going to have some number to the t power. I want to get as many things just raised to the t power as possible, just to see if I can simplify this thing. So this character right over here is going to be 81. Nine squared is 81. Five to the third power, 25 times five, that's 125. So we're making good progress and so the only thing we really have to simplify at this point is nine to the t over two. And actually let me do that over here. Nine to the t over two. Well that's the same thing as nine to the one half times t. And by this property right over here, that's the same thing as nine to the one, nine to the one half... And then that to the t power. So what's nine to the one half? Well that's three, so this is going to be equal to three to the t power. So this right over here is three to the t power. So now this is getting interesting. So I have the 10 out front, times three to the t, Actually let me write the 81 first. 10 times 81 times three to the t, so all I did is I just swapped the order that I'm multiplying. Times 125 to the t. Times 125 to the t power. Now, the 10 times 81, I can just multiply that out. That's going to be 810. And then what's three to the t times 125 to the t? Well this is another exponent property at play here. Because if I have a times b to the t power, that's a to the t times be to the t. Or another way to think about it, if I have a to the t times b to the t, that's the same thing as a times b to the t power. And so over here I have three to the t times 125 to the t, so it's going to be the same thing as three times 125 to the t power. So this part of it right over here, I could rewrite it as three times 125, and I'm gonna raise that whole thing to the t power. So I'm in the home stretch. This is going to be 810, times three times 125 is 375. Times 375, to the t power. And that's about as simplified as we can get, and we did indeed write it in the form that we hope to write it in. That a times be to the t. Where if we actually match this form right over here, this right over here would be our A, and then our B would be the 375. So anyway, I know it looked a little bit daunting when you first saw it, but if you just keep saying okay let's just keep applying some exponent properties here, let's see if I can get multiple things to the t power, and keep breaking it down using these properties, you see that in not too many steps you get to something that's a lot less hairy.