simplify i38.
Tuesday, February 19, 2019 11:44:10 AM
Dean

Now let's say we try something a little bit stranger. And then you have that 1 left over. And this is the same thing, just from our exponent properties, as i to the fourth power raised to the 25th power. And you might see these surface. So i to any multiple of 4-- let me write this generally. Let's take i to the 38th power. Sorry, that's not i to the i-th power.

And so this is going to be i to the 7,320 times i to the first power. And they're also kind of fun to do to realize that you can use the fact that the powers of i cycle through these values. This is i to the first power. And so this part right over here is going to simplify to 1, and we're just going to be left with i to the first power, or just i. So you have negative 1 times i is equal to negative i. Or if you forget that, you could just say, look, this is the same thing as i squared times i.

So you could say that this is the same thing as i to the 4 times 25th power. And so we are just left with i to the first. So that's just 1 to the 16th, so this is just 1. So you can't just do that that simply. This simplifies to 1, and I'm just left with i squared, which is equal to negative 1. So once again, what's the highest multiple of 4 that is less than 99? If you have something raised to an exponent, and then that is raised to an exponent, that's the same thing as multiplying the two exponents.

So if we have i to any multiple of 4, right over here, we are going to get 1, because this is the same thing as i to the fourth power to the k-th power. So this is going to be equal to i. And so you could rewrite this. Now, we just have to figure out this is going to be some multiple of 4 plus something else. So this is equal to 1 to the 25th power, which is just equal to 1. So let's try that with a few more problems, just to make it clear that you can do really, really arbitrarily crazy things.

I'm doing i to the 36th power, since that's the largest multiple of 4 that goes into 38. What's left over is this 2. When you multiply, you can add exponents. Let's try i to the 501st power. And if we have anything else-- if we have i to the 4k plus 1 power, i to the 4k plus 2 power, we can then just do this technique right over here.

So let's try, just for fun, let's see what i to the 100th power is. This is equal to i squared times i. So this would be i to the 501st power. And that is the same thing as 1 to the k-th power, which is clearly equal to 1. If you multiply these, same base, add the exponent, you would get i to the 99th power. Well, once again, this is equal to i to the 36th times i squared.

And then that times i to the first power. Well, i to the fourth is 1. So it seems like a really daunting problem, something that you would have to sit and do all day, but you can use this cycling to realize look, i to the 500th is just going to be 1. And we know that i to the fourth, that's pretty straightforward. You have the same base.

This is a multiple of 4-- this right here is a multiple of 4-- and I know that because any 1,000 is multiple of 4, any 100 is a multiple of 4, and then 20 is a multiple of 4. And then one that isn't. . And so i to the 501th is just going to be i times that. You can use this to really, on a back of an envelope, take arbitrarily high powers of i. So if you have i to any multiple of 4, so this right over here is-- well, we'll just restrict k to be non-negative right now. Let me do one more just for the fun of it.

And the realization here is that 100 is a multiple of 4. And we know that this is the same thing as-- i to the 500th power is the same thing as i to the fourth power. Now in this situation, 501, it's not a multiple of 4. You can verify that by hand. So that's this part right over here.

But what you could do, is you could write this as a product of two numbers, one that is i to a multiple of fourth power. And then you're just left with i to the third power. And so this right over here will simplify to 1. So you could write this as i to the 500th power times i to the first power. So this is the same thing as i to the 96th power times i to the third power, right? And you could either remember that i to the third power is equal to-- you can just remember that it's equal to negative i.