in-38-45-find-the-multiplicative-inverse-of-each-of-the-following-in-ab-i-form-frac563-i

Question

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

Jeremy Strong · Accepted Answer

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&#x27;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&#x27;s 56 ministry asked when he was six Mon Astrea and then in the denominator, we&#x27;re going to need to use the foil method. So we&#x27;re going to have 56 times 56 which is 25 over 36 and then we&#x27;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&#x27;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&#x27;s going to be 25 over 36. Now we&#x27;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&#x27;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&#x27;s gonna give us 30 over 349 as our real component, and then it is going to give us Let&#x27;s see, for our imaginary component, we&#x27;re gonna have modest three out Tom&#x27;s 36 that&#x27;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.

Heather Zimmers · Answer

the multiplication of inverse of three minus three. I would be 1/3, minus three I. But that&#x27;s not the form were supposed to leave our answer in. We want to put it in standard form. And to get there, we need to multiply by a special form of one the congregate of the denominator three plus three I over itself that will help the denominators simplify. So our numerator is three plus three I and then the denominators We need to multiply together. So if we foil those, we got three plus 39 And then the outsides make plus nine I and the insides make minus nine I. So those councils cancel and then the last give us minus nine. I squared. Okay, remember that I squared is negative one. So we have three plus three I over nine, minus nine times negative one. That means we have three plus three I over nine plus 9 18 and noticed that we could factor three out of the numerator and reduce that with the 18 on the bottom. And I gives us 1/6. So we have one, plus I over six. Now, if we want to split that up into two separate parts, the rial part would be 1/6 and the imaginary part would be 1/6 I.

Jeremy Strong · Answer

in this problem were asked to find the multiplication in verse of the complex number one plus I. So to begin, we need to rewrite one plus I as one over one plus I and this, technically speaking, is the complex number is the multiplicity of inverse. However, we need the answer to be in the form A plus B A. This is the form that we need. So how do we get one over one, plus I to the A plus b ah form. We need to multiply the numerator and the denominator by the complex congregate of the denominator. So we have one plus us. So the complex conjugal of that is one monitor. So we have to multiple the numerator and the denominator by one modest on. Okay, so in the numerator we have one. Tom&#x27;s one might decide to give us one minus I and then into the denominator. We actually need to foil. So we&#x27;re going to have one times one which is one and then our outer is Monness. Ah, And then our inner is plus, uh and then our our our last excuse me is positive on Tom&#x27;s Mon Issa, Theo&#x27;s modest squared and modest. I swear it is the same thing as a plus one. So now we have one minus five, divided by one monness. I plus, uh, plus one so modest. And plus I cancel. And that leaves us with one plus one, which is going to leave us a two in the denominator. So we have one minus. I divide about you now as our last step. We need to split this fraction apart into two separate components because we need this to be in the form a plus beyond. So we have one modest I over too. And that is equal to 1/2 modest 1/2 I and that is final.

Heather Zimmers · Answer

Let&#x27;s find the multiplicity of inverse of two plus four I. So that&#x27;s going to be one divided by two plus four I. But that&#x27;s not the form we&#x27;re supposed to give our answering. We want to give our answer and standard form. And to do that we need to get rid of the complex number in the denominator. So we&#x27;re going to multiply the numerator and denominator by two minus four I the contra git of two plus four I that will help us simplify the denominator. So when we multiply the denominators, we get four from multiplying the first for the outsides. We get negative eight I and for the insides we get positive. Eight I so that cancels. And then for the last we get minus 16 I squared and the numerator is just to minus four I. So let&#x27;s keep simplifying The denominator. I squared is negative one. So now we have two minus four I over four, minus 16 times. Negative one. So that&#x27;s two minus four I over four plus 16. So that&#x27;s two minus four. I over 20. Now let&#x27;s break that down into to over 20 minus four. I over 20 and simplified fraction and we get 1/10 minus 1/5. I

Jeremy Strong · Answer

in this problem were asked to find the multiplication of inverse of the complex number Negative one plus two. Uh, so but simply, this is simply one divided by negative one plus two. Uh, however, we need our answer to be in the form A plus B, I it must be in this form to be correct. So how do we simplify one over negative one plus two up to get it to April? Speak out for me. Well, we need to multiply the numerator and the denominator of this fraction by the complex continent of the denominator. So we will need to multiply the numerator and denominator about negative one Monness too, huh? I have one minus two, huh? So, in the numerator, we have one Tom&#x27;s negative one minus two. Uh, and that is simply negative. One minus, too. And the numerator and denominator, we&#x27;re gonna have to perform the foil method. So we&#x27;re going to get negative one time saying you want to give us one. Our outer were modest too. At times. Negative one is plus two, uh, our inner positive to ah, negative one. Multiplying them is going to be modest to, uh and then our last terms are going to be plus two out times Negative two out to give us monness modest for, uh, squared. Okay, moving over here. We still have negative one modest to ah, is our numerator divided by we have one plus two. I mon ist too I And then we had four. We had negative for r squared over here. Now I squared is the same thing as negative one. So instead of having negative for r squared, we&#x27;re going to rewrite that as a plus. Plus four. Okay, now we can re rot our denominator. So we have one plus one plus four is gonna give us five, and then we have plus two, uh, modest er which is going to cancel the of zero. So what we really have is we have negative one modest to, uh, about about five now, since we need our final answer to be in the form A plus beyond, we need to split this fraction into two separate fractions. So negative one modest to ah divided by five is equal to negative 1/5 modest to fifth. And that is the final answer.

Heather Zimmers · Answer

the multiplication of inverse of two minus 1/2. I would be 1/2 minus 1/2 I. But that&#x27;s not the form We want to leave our answer in. And in order to change the form, we&#x27;re going to have to multiply by a special form of one the congregate of the denominator, which is two plus 1/2 I over itself. Okay, so what does that give us? The numerator is two plus 1/2 I. And then to get the denominator, we can multiply by using the foil method. The first give us for the outside gives us one I The inside gives us negative one I. So those cancel and the last gives us negative. 1/4 I squared. Okay, lets keep going and simplify this. Remember that I squared is negative one. So we have two plus 1/2 I divided by four minus 1/4 times. Negative one. That&#x27;s the same as four plus 1/4. So we have two plus I two plus 1/2 i over four plus 1/4 Now, with all these fractions involved, it would be really handy just to multiply the top and the bottom by four and clear the fractions. So that gives us Don&#x27;t forget to distribute. So that gives us eight plus two I on the top, and that gives us 16 plus one on the bottom. So what we have now is a plus two. I over 17. Now, if you like to write that as two separate parts, the rial part would be eight seventeenth&#x27;s, and the imaginary part would be to 17th I.

Problem

In $38-45,$ find the multiplicative inverse of ea…

Problem 44 Hard Difficulty

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

Answer

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

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Ann Xavier Gantert

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Absolute Value - Example 1

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

Algebra 2 and Trigonometry

Grace H.

Alayna H.

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Absolute Value - Example 1

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Ann Xavier GantertAlgebra 2 and Trigonometry

Grace H.

Alayna H.

Kayleah T.

Kristen K.

Ann Xavier Gantert

Algebra 2 and Trigonometry