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In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.$$\frac{5}{6}+3 i$$

$\frac{30-108 i}{349}$

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 5

Operations with Complex Numbers

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

01:30

In $38-45,$ find the multi…

02:07

01:35

02:37

01:55

04:04

01:58

00:22

Find the additive inverse …

00:23

Find the inverse of each r…

00:56

In Exercises $35-42,$ find…

for this problem were asked to find the multiplication in verse of the complex number 56 plus three on. So moving right along, we may rewrite the multiplex of inverse as one divided by 56 plus three. And while this is technically the correct answer, we need our final answer to be in the form a plus B on in order for us to be correct. So what we're going to need to do is to multiply the top and bottom of this complex fraction by the complex contra kit of the denominator. Okay, so moving right along, we only the multiple the top in the bond about 56 Monastery, huh? So one Tom's 56 ministry asked when he was six Mon Astrea and then in the denominator, we're going to need to use the foil method. So we're going to have 56 times 56 which is 25 over 36 and then we're gonna have minus three times 56 to give us minus 15 over six I and then our winner is going to give us plus 15 for six I, and then our last is gonna give us modest nine I squared. Okay, moving right along. We still have 56 Monastery A and the numerator now in the denominator are modest. Not all square is going to cancel to give us a plus non instead of a modest none because I squared is equal to negative one so modest on Tom's minus one is gonna give us plus on. So our real component is 25 or 36 Plus, not when our imaginary component is negative. 15 over six plus 15 over six I and that actually is going to cancel out to give us zero. So the only thing remaining and the denominator is 25 or 36 plus none. So if we are good with multiplying by fractions and funding, um, common denominators, then we can multiply the pause. None by 36. So our denominator it's going to be 25 over 36. Now we're gonna have to multiply non by 36 that is going to give us 300 324 over 36. And it would be completely acceptable to use a calculator to do this 25 or 36 plus nine. But I'm just gonna show you how we would do it using fractions. So this is our new denominator. And in the numerator we have of six monastery, huh? Okay, Now we are getting down to the nitty gritty. So in the denominator, we have 25 or 36 plus 324 over 36 that works out to 349 over 36. And then we still have 56 minus three hours in the numerator. Now, at this point, we technically can split the fraction into the final answer, and it would give us above six, divided by 349 divided about 36 then mon ists three ah, divided by 349. You thought about 36? All right, Now we can go one step further, even though this is technically the correct answer. And we can multiply the numerator and the denominator by a 36 to get this fraction out of the denominator, and I will show that right here. So if we multiply the top of the bottom about 36 it's gonna give us 30 over 349 as our real component, and then it is going to give us Let's see, for our imaginary component, we're gonna have modest three out Tom's 36 that's gonna give us monos 108 over 349. Both of these answers are correct technically, but this is the simplified answer. 30 over 349 Monets, 100 Nate over 304 nuts.

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