00:01
To prove that the seventh root of seven is irrational, so let's assume it's not.
00:07
Assume not.
00:12
Assume the seventh root of seven equals p over q, where p and q are integers, and q is not zero.
00:26
Then let's raise both sides to the seventh power.
00:34
So we get seven equals p over q to the seventh.
00:41
7 equals p to the 7 over q to the 7th.
00:47
7 q to the 7 equals p to the 7.
00:53
Okay, that means 7 divides p to the 7th.
01:07
And so 7 divides p.
01:10
Okay, we might need some proof of that because 7 is prime.
01:17
So p equals 7m for some m in the, integers.
01:27
So 7q to the 7th equals p to the 7.
01:32
So 7 q to the 7 equals 7m to the 7th.
01:39
So 7 q to the 7 equals 7 to the 7, m to the 7.
01:46
So q to the 7 equals 7 to the 7 to the 6th, m to the 7th.
01:52
Oh, i forgot to say something.
01:56
Assume not.
01:57
Assume the 7th root of 7 is p over q.
02:00
P over q are integers.
02:02
Q isn't zero.
02:04
And p over q is in reduced form.
02:13
Okay.
02:14
If it's not in reduced form, then make it in reduced form...