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

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

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 5

Operations with Complex Numbers

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

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01:32

In mathematics, the absolu…

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In $46-60,$ write each quo…

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for this problem. We are dividing the complex Number four modest to, uh, into the complex number two plus two. Uh, so what we're going to need to do to simplify this complex fraction is we'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're gonna have to foil both the numerator and denominator. So to begin in the numerator, we're gonna have to Trump's for which is eight. And then we'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's our numerator. Same thing in the denominator. So we'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's gonna bust Monness eight. Uh, and then our last is modest to ah, Tom's positive Joe to give us modest four. Uh, squared. Now let'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'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'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.

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