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# Rewriting exponential expressions as A⋅Bᵗ

CCSS.Math:

## Video transcript

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 9 to the T over 2 plus 2 power times 5 to the 3t and what I want to 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 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 9 to the T over 2 plus 2 so this right over here I could break this up using the fact that I'll just write the properties over here if I have if I have 9 to the a plus B power this is the same thing as 9 to the a times 9 to the B power and over here I have 9 to the T plus T over 2 plus 2 so I could rewrite this as 9 to the T over 2 power times 9 squared times 9 squared all right now let's move over to 5 to the 3t well if I have a to the BC so you could view this as 5 to the 3 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 5 to the 3rd 5 to the 3rd 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 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 9 squared is 81 5 to the third power is 25 times five that's 125 so we're making a good progress and so the only thing we really have to simplify this point is 9 to the T over 2 and actually let me let me do that over here 9 to the T over 2 well that's the same thing that's the same thing as 9 to the 1/2 times T and by this property right over here that's the same thing as 9 to the 1 whoops 9 9 to the 1/2 times or 9 to the 1/2 and then that to the T power so what's 9 to the 1/2 well that's 3 so this is going to be equal to 3 to the T power so this right over here is 3 to the 3 to the T power so now this is getting interesting so I have the 10 out front I have the 10 out front times 3 to the t3 to the T actually let me write the 81 first 10 times 81 times 3 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 could just multiply that out that's going to be 810 and then what's 3 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 B to the T or another way to think about 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 3 to the T times 125 to the T so it's going to be the same thing as 3 times 125 to the T power so this part of it right over here I could rewrite it as 3 times 125 and I'm going to raise that whole thing to the T power so I'm in the homestretch this is going to be 810 times 3 times 125 is 375 times 375 to the 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 B 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 370 B the 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 not too many steps you get to something that's made a lot less hairy