00:01
Okay, this time we want to prove one of the exponent rules that if you have a fraction to the n power, you can make it the numerator to the n power over the denominator to the n power.
00:14
Since we're trying to prove it for all whole numbers greater than or equal 1, that tells me to use induction on it.
00:21
Prove by induction.
00:27
That means we have to first show that it works for n equals 1.
00:31
Second, we have to assume that it's true for n equals k, and then we have to show that it's true for k plus one.
00:40
And then by induction, it's always true.
00:42
Okay, so first step, show it is true for n equals 1.
00:53
That is show over b to the 1 equals a to the 1 over b to the 1.
01:05
Oh, that's not usually how i write it.
01:07
I just wasn't paying attention.
01:09
That's okay.
01:10
Okay, i'm going to start with the left -hand side.
01:12
A over b to the one.
01:14
Anything to the one powers, just the thing, because that's the definition of to -the -one power.
01:22
And a -to -the -one, b -to -the -one equals a -over -b.
01:27
So since a -over -b to the one equals a -over -b, and a -to -1 -b -to -1 equals a -over -b, then they must be equal to each other.
01:38
And that's the transitive property if you need to know.
01:43
So, a over b to the 1 equals a to the 1 over b to the 1.
01:48
So it's true for an equals 1.
01:52
Okay, second, assume it is true for n equals k.
02:04
That is, assume a over b to the k is equal to a to the k over b to the k.
02:18
Okay, now lots of times people try and do this without writing that step down.
02:24
I don't know why you would do that, but you need to write it down.
02:28
It makes it way easier.
02:30
Okay, once you do that, now show it is true for n equals k plus one...