In Exercises 53–60, write each power of i as as i, - 1, - i, or 1...

Question

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1.i^114

Accepted Answer

Welcome back. I'm so glad you're here. We've got this beautiful expression here. I raised to the 98th power and were to rewrite it as negative 11 negative I or I. Well what does that even mean? Well, we recall from previous lessons that I raised to the first power equals I. That I raised to the second power equals negative one. That I raised to the third power equals negative. I and that I raised to the fourth power equals positive one. Well, hey, those are four answer choices. How do we get from 98 to I. Negative I negative 11. How do we do that? Well, the key is this I to the fourth equals one. We want to figure out how many ones are in there or in other words how many I to the fourths are in I to the 98th. So what we do is we take that exponents that 98 and we divide it by four. Find out how many I to. The 4th star in there. So in I to the '98. There are 24. I the fourths. Well that's cool. So we can say we've got I to the fourth raised to the 24th and that gets us darn close to 98 but not all the way we're still missing a little bit. It's that 980.5. What does that 0.5 meat. Well, if we divide by four and we get a whole number, That means there's no remainder and the answer would have just been one. But if we get a remainder of .25, that is the same as saying we have a remainder of one or one eye left over. If we have a remainder .50, that's a remainder of two. And it means that we have two eyes left over and if we would have .75 as our remainder, that's a remainder of three, meaning three eyes left over. In this case we've got 0.5. So a remainder of two and we can say I squared or two eyes are left. Now let's look at our red chart and we can replace what we can I to the fourth. Well that's one. So we've got one raise to the 24th times I squared. Well that is negative one. So we've got one of the 24th. Well, one to whatever it's being raised to, that's just going to be one. So we've got one times negative one or -1. We look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1.i^114

Key Concepts

Powers of i

Modulus and Division

Complex Numbers

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