If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:8:32

Video transcript

but I hope to do in this video is start with the exponential expression that's in a fairly straightforward form and then turn it into one enough hairier form and let's actually just do it I'll show the initial expression in the form that I want it in or that we want it in and we could talk a little bit about why would we ever actually want to do that so let's say my expression is one thirty Seconds times two to the T power so this is fairly straightforward exponential expression but let's say we want it we want it in the form a times B and this is where it's going to get hairy eight times B to the T over ten power minus one and so you're probably immediately saying why would I ever want to take something nice and simple like this and turn it into this beastly thing right over here and the answer is when you get into higher math and you start doing your physics in your chemistry you're going to see maybe you know you got a result like this but then you look in your textbook or your professor it has a result like this just well how do I transfer from this to this or actually sometimes when you transfer to a form like this obviously I've just arbitrarily written this form here but sometimes when you Trant when you write it in another form that might even be a little hairier it can give you an intuition on the underlying processes that that expression is trying to describe so if you can take that on a leap of faith let's actually try to do it at minimum it's going to make you a lot better at exponent properties so see if you can rewrite this in this form so I'm assuming you took a go at it so let's try to do it together so the first thing that the first thing that I might want to do is well let's see if we can let's see what would I want to do the first thing I want to do is take this T and get it into a t let's get it into a t over 10 so to do that we essentially just need to multiply by 10 and divide by 10 so let's multiply by 10 and then also divide by 10 then we haven't changed the value up here so we can rewrite this we can rewrite this as 1/32 132nd times two times two to the so let me circle t let me just in a different color t t over 10 t over 10 times 10 x times 10 all right so we got a t over 10 over here but then i have this times 10 so how do i how do i deal with this well one thing that I could do let me just let me actually just write this the other way around let me write it as let me write it as 10 times T over 10 10 times T over 10 so hopefully this what I just did here isn't a huge stretch here I just literally multiplied and multiplied it and divided by divided by 10 so I have this T over 10 but when I write it this way an exponent property might jump out at you if I have if I have a to the B and then I raise that to the C that's going to be a to the B C or another way around a to the B C is going to be a to the B to the C and so this piece right over here I can rewrite it as 2 to the 10th and then raise that to the T over 10 power to the T over 10 power once again 2 to the 10th and then raise that to the T over 10 that's going to be the same thing as 2 to the 10 times T over 10 and of course we still have the 1 over 32 over here 1 over 1 over 32 and I'm tempted to write that is 2 to the negative fifth power but I won't do that just yet so let's see what's 2 to the 10th power actually let's just let's just keep it let's just keep it as 2 to the tenth power just for simplicity right now later we can you might know that that's going to be 1,024 but let's just let's see what else we can do so we know this is going to be some actually let me just write out as 1,024 so we have 1 over 32 times 1000 1,024 the T over 10 to the T over 10 power so it seems like we're getting close we did it we if there is no minus 1 here we're essentially done but now there's this minus 1 so how do we deal with that well we can do a similar type of strategy we can subtract 1 and then we could add and then we could add 1 then we're not actually changing the value just as we multiply it by 10 and divided by 10 we're not changing the value up here if you subtract 1 and add 1 to the exponent you're not changing its value and so what is this going to be we want to leave this minus 1 here but we want to get rid of we want to get rid of this plus 1 somehow and here we just have to remind ourselves that if we have a to the B times a to the C that's going to be equal to a to the B plus C if you have the same base multiplied to same base raised to different exponents and you multiply them you could just add the exponents and so you could also go the other way around if you have a to the B plus C you could break it up into a to the B times a to the C so this business right over here this business right over here this is 1024 to the T over 10 minus 1 plus 1 so we can break this up as we can break this up as 1024 1024 to the T over 10 minus 1 that's this part here and then x 1024 to the 1 times let me make this in a different color so let's see green so this right over here so x times 1024 to the 1 power that's this one power right over here and of course we still have the 1 over the 1 over 32 all right so now we're really close we have the 1024 to the T over 10 minus 1 with T over 10 minus 1 and now we just have to simplify we can rewrite this is going to be equal to this is going to be equal to we could just bring the 1024 1024 to the first power that's just 1024 so that's going to be 1024 over now I can put all these commas here if I like 1024 over or this 30 - let me do that magenta color over this 32 times homestretch times times 10 1024 to the T over to the T over 10 -1 power and now we could just simplify this you must you might recognize 1024 we already saw that was the same thing as 2 to the 10th power 32 is the same thing as 2 to the 5th power so 2 to the 10th divided by 2 to the 5th actually this is another exponent property at play here although you could just divide the numbers if you have if you have a to the B over a to the C this is going to be equal to a to the B minus C so this is going to be 2 to the 10 minus 5 or this whole thing this whole thing right over here and this thing see I wanted to do that in a different color this thing is just going to be 2 to the 5th power or 32 so this is going to be 32 times 1,024 one we were in the homestretch here 1024 to the T over 10 minus 1 so once again normally in our lives we like to make things simpler and I'm a big advocate of that it's a good life philosophy but this is a case where we really did make it more complicated we started with 1/32 times two to the T actually we could have eat well there's other ways we could have written that and we turned it into this thing with this somewhat hairier exponent but it's a useful skill to have because you might get a result like what we originally started with and then someone else might get a result like this and it's very important to realize hey you actually got the same result they're just different ways of expressing the same exponential expression