00:02
Okay, now they want us to calculate the powers of i from 1 to 12.
00:07
Let's start doing that because they also said they want us to find a formula.
00:11
So i to the 1 is just i.
00:15
I squared.
00:17
Well, we know that's negative 1.
00:19
Well, what's i cubed? that's just i squared times i.
00:24
But what's i squared? we can see it right here.
00:26
It's negative 1.
00:27
So it's negative 1 times i.
00:29
We can just write that as negative i.
00:31
What's i for well we return to negative i times i because it's i cubed times i so let's just write that out i cubed times i but we just saw that i cubed is negative i so we have negative i times i which is negative i squared but i squared is just negative one so we have negative negative one which is just one so we saw that i to the 1 is equal to i, i to the 2 is equal to negative 1, i to the 3 is equal to negative i, and i to the 4th is equal to 1.
01:22
Now we're going to start seeing something pretty cool, because what is i to the 5th? well, that's just equal to i to the 4th times i, which is just...
01:39
I to the fourth times i.
01:42
That is just 1 times i, as we just saw.
01:46
So it's just i.
01:48
So we can write over here, i to the fifth is equal to i as one.
01:56
Well, what's i to the six? it's i to the fifth times i.
02:00
And that is just i times i.
02:04
So that's the same thing as i squared.
02:07
So we can write over here, i6 is also equal to negative one.
02:13
What about i7? that's just equal to i to the 6 times i.
02:23
But what's that? well, that's just i squared times i, which is i cubed.
02:31
So i to the 7 is just equal to i cubed.
02:37
Now, you should start thinking, hey, what's going on here? we're just repeating ourselves.
02:44
And that's perfectly right, because if we try to compete, i to the eighth now.
02:49
We'll just end up with i to the seven times i, but we know what i to the seven is.
02:58
It's the same thing as i to the three times i, which is i to the fourth.
03:04
And we know what that is from before we computed it.
03:07
It's i to the one.
03:12
It's i to the fourth, which is one.
03:14
Okay.
03:16
So we're starting to see this really nice repetition right here.
03:22
Let's see it one more time and come to a conclusion.
03:27
I to the 9th.
03:29
Well, that's just i to the 8th times i.
03:36
But what was i to the 8th? i to the 8th was just i to the 4th, which is just 1.
03:45
So we just have i.
03:48
We have that i to the 10th is equal to i to the 9th times i.
03:55
Well, we know that i to the ninth is just equal to i.
03:59
So we have i squared here, which is negative 1, i to the 11th, that's equal to i to the 10th times i.
04:09
We just found out what that was...