in-46-60-write-each-quotient-in-ab-i-form-frac22-i4-2-i

Question

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{2+2 i}{4-2 i}
$$

Jeremy Strong · Accepted Answer

for this problem. We are dividing the complex Number four modest to, uh, into the complex number two plus two. Uh, so what we&#x27;re going to need to do to simplify this complex fraction is we&#x27;re going to need to multiply both the numerator and the denominator by the complex con jacket of the denominator. So the complex contradict of four minus two, uh, ISS for plus two. Uh, and we have to multiple of both the numerator and the denominator by four plus two. Uh, so that being said, we&#x27;re gonna have to foil both the numerator and denominator. So to begin in the numerator, we&#x27;re gonna have to Trump&#x27;s for which is eight. And then we&#x27;re gonna have two times two eye, which is a plus for, uh, in our interests to odd times four, which is gonna give us plus eight. Uh, and then our last two out times too. I was gonna give us plus four squared. OK, so that&#x27;s our numerator. Same thing in the denominator. So we&#x27;re going to do four times four 16 then our outer is four times plus two oddity of us plus eight. Uh, our interest modesta time source. That&#x27;s gonna bust Monness eight. Uh, and then our last is modest to ah, Tom&#x27;s positive Joe to give us modest four. Uh, squared. Now let&#x27;s look at these. Modest these Excuse me. Ah, squared components. So ah, squared works out to be negative. One negative one because the square root of negative one squared cancels the square root to leave us with the negative one. So instead of having a plus for us squared, we&#x27;re going to have plus four times negative one to give us a modest four in the numerator. And then down here, we have modest for a squared, which is modest for Tom&#x27;s negative one, which all works out to be a plus. For now, in the numerator, we have eight minus four to give us four. And then we have four AA plus eight AA, which is a plus 12. I Okay, so our numerator has been simplified. Now for our denominator, we have poolside and modest eight I which cancel one another out. We have 16 plus four, which is 20 now. We need our final answer to be in the A plus B form for it to be correct. So we have to recap. We have four plus the 12. I divided by 20. And this is equal to four over 20 plus 12 over 20 I and this could be further simplified to 1/5. Pull us 3/5. Uh, and this is our final answer. 1/5 plus 3/5.

Heather Zimmers · Answer

First of all, let&#x27;s change this quotient into the following form two plus four I over one minus I and then to change it into standard form. What we can do is multiply the numerator and denominator by one plus I, which is the contra git of the denominator. This is going to help us to simplify the denominator. So we&#x27;re going to multiply the enumerators by using the foil method. Multiply the first, the outsides, the insides and the last, and we get to plus two I plus four I plus four. I squared and then we multiply the denominators in the same way. But because they&#x27;re congregates, the insides and outsides will cancel. So we&#x27;ll end up with one minus. I squared. Now we want to do some simplifying. So in our numerator we can add the two y and the four I and that will give us six I and also remember that I squared is negative one. So we have four times negative. One on the bottom will also replace the I squared with negative one. So we one minus negative one. So we can simplify this because we have to plus four times negative one that&#x27;s going to be to minus four. So that gives us negative two plus six. I over and the denominator will be, too. Now everything&#x27;s divisible by two so we can change this to negative one plus three I

Jeremy Strong · Answer

in this problem, we are asked to divide the complex number one plus on about the complex number 1/2 minus 1/2 up. So to begin, we&#x27;re going to need to multiply this entire fraction both the numerator and the denominator by the complex congregate of the denominator, which is going to be 1/2 plus 1/2 y 1/2 plus 1/2 y. Now we&#x27;re going to need to foil both the numerator and denominator. So, beginning in the numerator, we have one that Tom&#x27;s 1/2 which is 1/2 and our outer is one. And what have I So it&#x27;s gonna give us plus another what, half times I and then our inner is I want and what, half so yet again, we&#x27;re gonna have a plus 1/2 and then our last times are going to be a plus. 1/2 squared, all right. And that is our numerator now doing the same thing in the denominator. We&#x27;re gonna have 1/2 times 1/2 which is 1/4 Now we&#x27;re gonna have 1/2 times 1/2 I, which is going to be plus 1/4 and then our inner components are modest 1/2. Ah, and what happened? So it&#x27;s gonna give us modest 1/4 I and our last components modest one F and 1/2 I we&#x27;re going to combine to give us monos 1/4 squared. Now let&#x27;s take a look at these are squared components. So I squared is the same thing as minus one because that is the square root of minus one squared, which leaves with a minus one. Now in the numerator, we have 1/2 times negative one, and that is going to leave us with a monness. 1/2 in the numerator lock wasn&#x27;t the nominee. We have modest 1/4 times a minus one, which is gonna work out too, plus 1/4. Okay, Now, in the numerator, we have 1/2 modest 1/2 and that is going to leave us with a real component of zero. So our imagine it the minute we have a positive 1/2 I am positive 1/2 I that&#x27;s going to be plus one eye. So that is our new numerator. And then in the denominator, we have 1/4 plus one force elects to fourths and then we have plus 1/4 and modest 1/4 eye, and that&#x27;s going to cancel. So our new okay, fraction rerouting it over here is simply ah, divided by two quarts. Okay, now we can simplify this just a little bit more because it is important to knowledge. We are missing a real component. We can go ahead and rot a zero, but it&#x27;s entirely necessary. Zero plus. So we have odd about about 2/4. So we may simplify this by multiplying the top and the bottom of the fraction before, and it will cancel this four and the denominator which will leave us with zero plus four, uh, divided about you. And then we can further simplify this down to zero plus for over two and four out. About about two. It&#x27;s simply too Ah, and that is the final.

Jeremy Strong · Answer

in this problem, we&#x27;re has to divide the complex number eight plus six. I buy the complex number two up now. Ordinarily, we would need to multiply the top and the bottom of this fraction by the complex conjugal of the denominator. And we can still do that, Rob at the final answer. But it&#x27;s not entirely necessary here because since there is just one imaginary component and the denominator and there is no real component, we can already split this fraction into two separate fractions. So we have a pool six are divided by two out, which is the same thing is eight divided about to, uh, plus six I divide about You are so in the six out, about about two out, the odds were going to cancel. And sixth, about about two is gonna live us with a three. Now we still have three plus eight divided by two I. This is not in the proper form because we need our answer to be in the form A plus B A for it to be correct. So how can we get this eight to ah, here To be in the proper form? Well, we will need to multiply the top and bottom of the fraction by the complex continent of the denominator, which is negative to ER, which we could have done at the start of the problem. But it we could do the simplification to go ahead and get the three before doing the multiplication by the complex conjugal. So now we have three plus our eight divide about to ah, and we needed multiple of the top and the bottom by negative to I which is the congregate off too. All right, now we have three plus that we have eight times negative to ER which is negative 69. And then in the denominator, we have negative to ah times to er which is negative for Ah, squared. Now squared is the same thing. It&#x27;s negative one because the square root of negative one squared is negative one so negative four times this negative one is all going to work out to be a positive for all right. And our final answer is we have three plus negative 16. I devote about four, and this simplifies to three minus four. Ah, and this is our final and are correct

Jeremy Strong · Answer

in this problem, we are asked to divide the complex number eight plus two Ah, by the complex number one plus three up. So to simplify this complex fraction, we will need to multiply both the numerator and the denominator by the complex conjugate of the denominator. The purpose of doing this is so that our denominated will cancel out to be a just a real number without an imaginary component. So the complex conjugal of the one plus three eyes one monastery I and we have two most while this body numerator and the denominator. So we&#x27;re gonna have to foil in both the numerator denominator. So to begin, in the numerator, we have eight times one which is eight. Our outer components are eight and monastery out to give us Mona&#x27;s 20 fora. And our inner components are two and one, which we&#x27;re going to give us plus two. Ah, and then lastly, our last components are plus two and monastery on, which is gonna give us monos six squared for our numerator. Now in the denominator, we need to do the same thing. So we have one times one to give us one hour. Outer is monastery out Tom&#x27;s one, which is minus three. Ah, our interest plus three out Tom&#x27;s one Nicholas plus three hours and then our last. It&#x27;s plus three times of Monastery on to give us a modest nine squared. It&#x27;s okay now, to simplify a little bit. We have These are squares here and square. It is the square root of negative one squared. So each one of these actually a negative one. Now we have minus six. Tom&#x27;s the negative one, and that is going to work out to be a plus six and then in the denominator, we have modest nine times. Negative one which works out to be a plus, not now. Moving right along. We have eight plus six in the new waiter, which is gonna give us 14. And then we have modest 24 AA plus Chua, which is gonna give us Mona&#x27;s 22 for our simplified numerator and then in the denominator, we have one hawass non to give us 10 and then modest area. Plus I cancels. So we just have a tent in our denominator. Moving right along. We have 14 monos 22 all divided by 10 but to get our final answer in the A plus B, uh, form. We need to split this fraction apart. So this is actually equal to 14. Over. Chen Monos 22 over 10. Ah, and this could be further simplified to seven over five. Modest 11 over five. Ah, and this is our final incorrect.

Heather Zimmers · Answer

all right, so we want to rewrite this quotient in standard form. And the first thing I would see is that we could factor of five out of the top five times a quantity to minus I, and we can factor two out of the bottom two times a quantity one plus three I and that will make the next steps a little bit easier. So the next thing we want to do is multiplied by a special form of one. The conjugated of one plus three. I that would be one minus three I because that&#x27;s going to simplify the denominator quite a bit. So here&#x27;s what we have. Now we have the five over to We&#x27;ll just leave that out and then we&#x27;re going to multiply using the foil technique first outside, inside last, and we end up with two minus six. I minus I plus three. I squared over same thing on the bottom. First outside, inside last. But because we&#x27;re multiplying congregates the outsides and insides will cancel. So first will be one and then last will be minus nine. I squared. Okay. There&#x27;s a lot we can do to simplify. First of all, we have negative. Six i and negative one. I will add those together and get negative. Seven I and every time we have a nice squared will replace it with negative one. So here&#x27;s what we have now. Five halves times Tu minus seven I plus three times negative. One over one, minus nine times negative one. Let&#x27;s simplify that a little more. We have five halves times Tu minus seven. I minus three over one plus nine and then let&#x27;s combine like terms two and negative three. So we have five halves times negative. One minus seven I over 10. Okay, let&#x27;s go ahead and reduce the five and 10. That gives us a two on the bottom in a one on the top. So what we have at this point is negative one minus seven I over four. And then if we want to break that down into two separate parts, the rial part would be negative. 1/4 and the imaginary part would be plus 7/4. I

Problem

In $46-60,$ write each quotient in $a+b i$ form. …

Problem 50 Hard Difficulty

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{2+2 i}{4-2 i}
$$

Answer

$\frac{1}{5}+\frac{3}{5} i$

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Ann Xavier Gantert

Algebra 2 and Trigonometry

Catherine R.

Heather Z.

Alayna H.

Caleb E.

Absolute Value - Example 1

Absolute Value - Example 2

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$\frac{1}{5}+\frac{3}{5} i$

Algebra 2 and Trigonometry

Catherine R.

Heather Z.

Alayna H.

Caleb E.

Absolute Value - Example 1

Absolute Value - Example 2

Ann Xavier GantertAlgebra 2 and Trigonometry

Catherine R.

Heather Z.

Alayna H.

Caleb E.

Ann Xavier Gantert

Algebra 2 and Trigonometry