Hey, everyone. We now know that *i*, our imaginary unit, is equal to the square root of negative one, which is great, but many problems are actually going to take *i* and raise it to a power. So *i* may be raised to the second power, the third power, or even something much higher, like *i* to the 100th power. Now I know that looks a little scary to calculate now, but I'm going to walk you through some lower powers of *i* that will then allow us to calculate these much higher powers of *i* super easily. So let's get started. Now, just like we are able to use all of our radical rules for the square roots of negative numbers, we're able to use all of our properties of exponents with our powers of *i*. So let's go ahead and look at our first power of *i*, *i* to the first power.

So if I raise anything to the power of 1, it's just going to be itself, and it's no different here. So *i* to the first power is just *i*. Now if I take *i* squared, if I take *i* and I square it, I know that *i* is just the square root of negative one. So if I take that and I square it, I know that squaring a square root just cancels it, and I'm going to be left with negative one. *i* to the third power, I can go ahead and use some other properties of exponents here and simply expand this into being *i* squared times *i* to the first. Now we just calculated both of those powers, so I can just take what I already know those are and plug them in. So *i* squared, we know, is negative one, and then *i* to the first power is just *i*. So I get negative one times *i*, which is just negative *i*.

Now *i* to the 4th power, I can just take *i* squared and multiply it by *i* squared because that is equivalent to *i* to the 4th power. Now I know that *i* squared is just negative one. So this is just negative one times negative one, which gives me positive one. So that's *i* to the first is just *i*, *i* to the second is negative one, and so on. Let's go ahead and move on to *i* to the 5th power. Now *i* to the 5th power, I can expand this out into being *i* to the 4th times *i* to the first power. *i* to the 4th power is a great thing to simplify these down into because since it's just one, it's going to make these really easy. So *i* to the 4th power again is just 1, and then *i* to the first is again *i*. I'm left with 1 times *i* which gives me *i*.

Now you might notice that *i* to the 5th power is the exact same thing we got for *i* to the first power. So let's see what we get for *i* to the 6th. So *i* to the 6th, I can expand that into *i* to the 4th times *i* squared. So *i* to the 4th, again, is just 1, and then *i* squared is a negative one. 1 times negative one gives me negative one. And this negative one that I got for *i* to the 6th power is the same thing that I got for *i* to the second power. So you might notice a pattern here. What do you think *i* to the 7th will be? Well, it'll actually just be negative *i*, the exact same thing that *i* to the third power was. So *i* to the 8th power is, you guessed it, just one, the exact same thing that *i* to the 4th power was. So this pattern is actually just going to repeat itself over and over again here. So if I were to take *i* to the 9th power, that would just restart the pattern. I would get *i*, *i* to the 10th power would just be negative one and so on. So that means that every single power of *i*, no matter how high it is, a 100, a 1000, a million, can always be simplified to one of these four values: *i*, negative 1, negative *i*, or 1. That's all for this one. I'll see you in the next one.