Hey, everyone. In the previous video, we learned how to calculate powers of i, and we saw the powers of i just cycle through 4 possible values, i, negative one, negative i, and 1. So we know that any of our powers of i are always going to be one of these four values. But if I'm given something like i to the power of 100, I don't really want to have to count through that cycle until I get to the i to the 100th power. So I'm actually going to show you a much quicker way to evaluate these higher powers of i. Let's go ahead and get started.

So we can actually express all powers of i in terms of i to the 4th power. Now this is going to be really useful for us because i to the 4th power we know is just 1. So if I'm given something like i to the power of 20, I can go ahead and expand this out and express it in terms of i to the 4th power by taking i to the 4th power and multiplying it by itself 5 times. Now since i to the 4th power is just one, this is the same thing as 1 times 1 times 1 times 1 times 1, which we know is just 1. So i to the 20th power is really just 1. But what if I'm given something like i to the power of 22? How might I express that in terms of i to the 4th?

Well, I can still start with my 5 powers or my 5 terms of i to the 4th power, and that gets me all the way up to i to 20th, but I still have 2 left over. So I have all of these i to the 4th terms, which I know just give me 1, but then I have this i squared term. So this is really just 1 times i squared. Now, from our previous video, we know that i squared is just negative one. So this is really just 1 times negative one. Now, 1 times negative one is just negative one. So i to the 22nd power is equal to negative one.

But still, if I'm given i to the 100th power, I still don't really want to have to write out i to the 4th power as many times as it takes me to get there. So I'm going to show you an even quicker shortcut for calculating higher powers of i. So to evaluate i raised to a very high power, we're going to ask ourselves one question, and that is, is our power evenly divisible by 4? Now if my answer to that question is yes, then I'm in luck because I'm done and my answer is just 1. So if I'm given something like i to the power of 100 and I ask myself, is this power evenly divisible by 4, my answer is yes because 100 divides by 4 25 times. So I know that my answer is just 1. i to the power of 100 is equal to 1. But if my answer is no, I go ahead and divide by 4. So 4 is still super important here, and we're going to use it to find our answer by dividing our power by 4.

Let's take a look at that in action. Now, if I'm given i to the power of 22 and I ask myself, is this power evenly divisible by 4? My answer is no. So I need to go ahead and divide this by 4 and find my remainder. Now if I take 22 and I divide it by 4, 4 goes into 22 5 times. 4 times 5 is 20, and I'm left with 2. Now 4 can't go into 2 at all, so 2 is going to be my remainder. This tells me that i to the 22nd power is going to be equal to i to the power of my remainder 2. Now, again, we know that i squared is just negative one. So my answer is negative one. This is the same answer that we got by expanding it in powers of i to the 4th by simply dividing by 4 and looking at our remainder.

Let's take a look at another example. So if I have i to the power of 67 and I ask myself, is this power evenly divisible by 4? My answer is again no. So that means that I need to go ahead and divide by 4 to get my remainder. So if I take 67 and divide it by 4 using long division, 4 goes into 6 one time, and I'm left with 2. If I bring down that 7, 4 goes into 27, 6 times. So 4 times 6 is 24, and that leaves me with 3 left over. And 4 doesn't go into 3 at all, so that means that my remainder is 3. So this i to the power of 67 is really just i to the power of 3. So in this case, my remainder is 3 and we know that i cubed is just negative i. So i to the power of 67 is really just negative i. So my remainder will always be either 1, 2, or 3, so I know that I can easily calculate those values to get my answer. That's all for this one. Let me know if you have questions.