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^135

Accepted Answer

Welcome back. I'm so glad you're here. We've got this beautiful expression here. I raised to the 115th power and were to rewrite that is negative 11 negative I or I? Well, what does that mean? Exactly? Well, we recall from previous lessons that when we have I raised to the first power that's I. We have I raised to the second power. That's negative one. We have I raised to the third power. That's negative. I and I raised to the fourth power is one. Well those are our four answer choices negative 11 native I or I. But how do we get from 100 and 15 to any of those? Well, the key is this I to the fourth equals one. Because one times however many ones you've got is just going to be one. So we want to know how many ones are there in 100 I to the 115th or in other words how many I to the fourths are in I to the 115th. So we take 115 and divide it by four to see how many I to the fourth star in there. We do that in a calculator and we get an answer of 28.75. Well, cool, that means there are 28 I two, the fourths in I to the 115th. So I to the fourth raised to the 28th Power Will get us close to I to the 115th, but there's still something missing. There's this .75 left over to get us up to 115. What does that mean exactly? Well, we recall that if you have a whole number When you divide by four, there would be no remainder. And it would just mean that your answer is one. But if we have a .25 that will give us a remainder of 11 I left over. If we have a .50, that's a remainder of two or two eyes left over, We have a remainder .75. That's a remainder of three or three eyes left over. And that's what we have. In this case we have 28.75 or three eyes left over. Or I. To the third. Well now we've got that divide up and we can look at our red chart and we can say okay I to the fourth. Well that's just one raised to the 28th power and I to the third. Well that is negative. I one race to whatever power it's being raised to. That's just one. So one times negative i is negative. I we look at our answer choices and this matches with answer choice. C. Well done. We'll catch on the next one

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i^135

Key Concepts

Powers of i

Modulo Operation

Complex Numbers

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