In Exercises 95–99, perform the indicated operations and write th...

Question

In Exercises 95–99, perform the indicated operations and write the result in standard form.(i^98 - i^94)/i^49

Accepted Answer

Welcome back. I'm so glad you're here. We've got this expression we have I to the 126th minus I. To the 122nd. All divided by I raised to those 63rd. Well, we're going to recall a few things about the nature of I. In order to simplify this, we know that I raised to the first is I. We know that I squared is negative one. We know that I to the third Is negative I. And we know that I to the 4th equals one. And this idea of the fourth equals one is very helpful because what we want to find out is how many I to the fourths are there in each of these big exponents. So we're going to take these big exponents and we're going to divide them by four to find out how many I to the fourth there are. So we take 126 and divide it by four. And on your calculator you will get 31.50. I want to pause for a second and remind us that if you get a whole number, that means there's no remainder. If you get .25, That's a remainder of one, meaning there's one I left over. If you get a .50 That means there's a remainder of two or two eyes left over. And if you get a .75, that's a remainder of three or three eyes left over. So 26 divided by four, that's 31.5 or 31 with a remainder of two. We take 122 divided by four. How many I. To the fourth star in 122. Well there are 30 with again a remainder of two or 30. Out of the 63rd. How many out of the 4ths are in there? Well that's going to give us 15 I. To the fourths with a remainder of three or three eyes left over. So now let's rewrite that I to the 126 is I to the fourth. How many of them? There's 31 of them so I to the fourth, raised to the 31st with two eyes leftover or times and I squared minus how many I to the fourths in 122. Well there's 30 of those times. How many leftover? There's two left over. I squared divided by eye to the 63rd. How many I to the fourths are there in the 63rd there are 15 with three left over. So I to the third left over. Now we can go back to our red chart and we can replace our eye to the fourths get replaced with one. So we've got one to the 31st times I squared. Well that's negative one minus I. To the fourth again is one raised to the 30th times I squared which is still negative one All over I to the fourth which is one raised to the 15th times I to the third which is negative. I All right. Do a little more simplification. One raise to any exponent at all is going to just be one. So instead of one to the 31st that's just one. So we have one times negative one. Well that's negative one minus one of the thirties. That's one. So one times negative one is negative one All over one to the 15th. Well that's just one times negative. I gives us negative. I now in the numerator we've got negative one minus negative one or negative one plus one which is zero all over negative I and we know that zero divided by anything is zero which gives us our answer choice. B. Well done, we'll catch you on the next one.

In Exercises 95–99, perform the indicated operations and write the result in standard form.(i^98 - i^94)/i^49

Key Concepts

Complex Numbers

Powers of i

Standard Form of Complex Numbers

Watch next