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In $19-34,$ write each sum or difference in terms of $i$$$-\frac{1}{2}+\sqrt{-\frac{2}{3}}-\frac{1}{2}+\sqrt{-\frac{24}{9}}$$

$-1+i \sqrt{6}$

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 4

The Complex Numbers

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

Campbell University

Harvey Mudd College

Idaho State University

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Okay, so this one is gonna be a little more work here. So what I've got is one over seven, plus the square root of negative one over eight, plus two over seven, minus the square root of negative 1/2. So to start, I can combine these terms because their numbers and they also already have the same denominator. So that does make it a little bit easier here. So I have three over seven and then plus, and I'll take out that negative one. And then I have the square root of one over the square root of eight. And then I have are negative negatives. Quit the square root of negative one. And then I have the square root of one over the square root of two. So here are three over seven stays, as is for now. And then I have a plus. This is an eye. This becomes one over and we do have to find the perfect square. Here is the four times the two. And so then I get times the square root of two times the square root of four minus I. And here the square one is one. So I get the one over the square root of two. So three over seven stays, and then here I have our eye times one over, and then this is too. So it's two times the square root of two minus I one over square root of two. So generally we don't leave these radicals in the denominator. Right? So what you can do is you can either get this radical out right away. Or you can get these to have the same denominator first and then remove the radicals. Either way is really the same difference. So let's just do this here for So in order to get this denominator to be the same as this one here, I just multiplied by two over, too. Because when I multiplied by one, I'm not changing anything, right? So then I have and you can put this I up here, you can keep it separate. I just like it separate for now because it keeps things a little bit easier until I have to square root of two minus. And then here I have I to over two times the square root of two. So here, three over seven. And then here. You see that I have the same denominator so I can combine these terms and you see that this minus sign is on the bigger number here. So I have a minus one over two square root of two I. And then if we don't want to leave that and the denominator, all I do is multiply the top in the bottom by the square root of two. So I'm left with three sevens minus and I have a square root of two. And I can put that I up in front now and then a two times when I have two radicals times each other. That's just the number that's inside there. So this becomes four. So that is my final answer here.

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