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 gonna 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 1/100 power. Now, I know that looks a little bit scary to calculate now, but I'm gonna 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 were 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 one, 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 fourth power I can just take I squared and multiply it by I squared because that is equivalent to I to the fourth 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 fifth power. Now I to the fifth power, I can expand this out into being I to the fourth times I to the first power I to the fourth power is a great thing to simplify these down into because since it's one, it's gonna make these really easy. So I to the fourth power again is just one and then I to the first is, again, I, I'm left with one times I, which gives me, I now you might notice that I to the fifth 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 sixth. So I to the sixth, I can expand that into I to the fourth times I squared. So I to the fourth again is just one and then I squared is a negative 11 times negative one gives me negative one. And this negative one that I got for I to the sixth 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 seventh 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 eighth power is you guessed it just one, the exact same thing that I to the fourth 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 ninth 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? 100 1000 a million can always be simplified to one of these four values I negative, one negative I or one that's all for this one. I'll see you in the next one.
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Higher Powers of i
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Video transcript
Hey, everyone in the previous video, we learned how to calculate powers of I. And we saw the powers of I just cycle through four possible values. I negative, one negative I and one. 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 I to the 1/100 power. So I'm actually gonna 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 fourth power. Now this is gonna be really useful for us because I to the fourth power we know is just one. 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 fourth power by taking I to the fourth power and multiplying it by itself five times. Now, since I of the fourth power is just one. This is the same thing as one times, one times, one times, one times one, which we know is just one. So I to the 20th power is really just one. 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 fourth? Well, I can still start with my five powers or my five terms of I to the fourth power. And that gets me all the way up to I to the 20th. But I'm still have two left over. So I have all of these, I to the fourth terms which I know just give me one. But then I have this I squared term. So this is really just one times I squared. Now, from our previous video, we know that I squared is just negative one. So this is really just one times negative one. Now one 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 1/100 power, I still don't really want to have to write out I to the fourth power as many times as it takes me to get there. So I'm gonna show you an even quicker shortcut of calculating higher powers of I. So to evaluate I raised to a very high power, we're gonna ask ourselves a one question and that is, is our power evenly divisible by four. Now, if my answer to that question is yes, then I'm in luck because I'm done. And my answer is just one. So if I'm given something like I to the power of 100 I ask myself, is this power evenly divisible by four? My answer is yes, because 100 evenly divides by 4, 25 times. So I know that my answer is just one I to the power of 100 is equal to one. But if my answer is no, then I need to take an extra step. And my answer is not just one, it's I to the power of the remainder of dividing by four. So four is still super important here and we're gonna use it to find our answer by dividing our power by four. So 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 four, my answer is no. So I need to go ahead and divide this by four and find my remainder. Now, if I take 22 and I do divided by 44 goes into 22 5 times, four times five is 20 I'm left with two. Now, four can't go into two at all. So two 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 two. 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 fourth by simply dividing by four. 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 four? My answer is again, no. So that means that I need to go ahead and divide by four to get my remainder. So if I take 67 and divide it by four, using long division four goes into 61 time and I'm left with two. If I bring down that 74 goes into 27 6 times, so four times six is 24 and that leaves me with three left over and four doesn't go into three at all. So that means that my remainder is three. So this I to the power of 67 is really just I to the power of three. So in this case, my remainder is three and we know that I cube is just negative. I, so I to the power of 67 is really just negative. I, so my remainder will always be either 12 or three. 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.
