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Factor the expression completely. Begin by factoring out the lowest power of each common factor.$$x^{-1 / 2}(x+1)^{1 / 2}+x^{1 / 2}(x+1)^{-1 / 2}$$...

Question

Factor the expression completely. Begin by factoring out the lowest power of each common factor.$$x^{-1 / 2}(x+1)^{1 / 2}+x^{1 / 2}(x+1)^{-1 / 2}$$

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$x^{-1 / 2}(x+1)^{1 / 2}+x^{1 / 2}(x+1)^{-1 / 2}$$



Answers

Fractional Exponents Factor the expression completely. Begin by factoring out the lowest power of each common factor.
$\left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2}$

Question us most factor to given expression extraordinaire. 1/2 times experts one to the 1/2 plus x to bill on half times X plus one to the negative 1/2. And the question gives us a little hint to factor out common factors but taking the lowest exponents. So in this we can see that there's the accident Commons, where we have to use this X because it has the lower exponents expose. One is also kind between both. But we have to factor in this term X plus one to the night of 1/2. Since that has the lower exponents. So start by factoring goes out next to the negative one, huh? Times X plus one to the negative 1/2. And then we just have to see what gets left behind when we do that. So if in the first term we simply get an X plus one came from the second term, we dare Justin x No, we can't back throughout this second set any further, so we could just combine the like terms and get our final I answer next to the native 1/2 times thanks plus one to the native 1/2 times two x plus one. And that's the final answer

We have to factor the given expression one post one over X squared, minus one minus one over X squared. So this is a difference of squares, So we'll distract it according to the formula for that. So that one plus one over X plus one minus one over X in the first set of fantasies in the second set will start with one plus one over access again. But understand, we have to subtract one minus one over X, so we'll simplify this further by combining like terms. So in the first part, we get one plus one, which is to one of our X minus one over X. That's just zero. So we just have to in the first parentheses, get one, and this negative sign is going to distribute over, actually, So this will become a negative one, and this will become a positive one over X. So we got one minus one, which is just zero. Then you get one over X plus one over X, which is just too over X. You can simplify this even further just by multiplying this out two times. Two over X is simply just four over X, and that is our final answer

Were factory this expression completely. And I'm going to call this part here the first term and this part here the second term. So looking at both terms, what do they both have in common that we could factor out? So what is the greatest common factor? They both have an X minus one, so we could factor that out. And they both have an X Plus two, so we could factor that out. Now, what is left in the first factor? We factored out the X minus one. We factored out one of the X Plus two. So there is still in X plus two remaining and what is left in the second factor. We factored out one of the X minus ones, so there is still an X minus one remaining, and we factored out the X Plus two. So here's what we have now, so we can simplify that third factor. We have our first factor X minus one, our second factor, X Plus two and our third factor. Let's go ahead and distribute the negative signs. We get X Plus two minus X plus one noticed the X's cancel. So we're left with just three. So we have three times x minus one times X plus two

You have to factor to. Given expression X plus one few times X minus two Experts one square times x squared plus X c eight times explosive. To simplify this a little bit, we're going at first find the Grace Kahn factor. Infected that out Now include this sex and the sex was went. So a factor these two parts out and then we'll try to simplify further. From there, they got X times X plus line, and then we're going to see what got left the height. So from this first term, we still haven't exposed one squared a minus two x times X plus one plus X squared. So there is a lot going on in this arm parentheses, and there's no real way to factor it out the way it is. So what I'm gonna do is I'm gonna just multiply it out and see if we can simplify it a bit more from there. So you still get our X Times X plus one and I'm gonna multiply this out. Sex plus one squared becomes X squared plus two x plus one. And I know this because that is a perfect square. No, we have to multiply this negative two x across the X plus one sending of to extend sex is negative. Two X squared intended to extents one is also never have two X and finally, our plus X squared thumb. Here we can just combine the like terms so we'll still have our X Times X plus one. But we should be able to simplify that second part a lot for a donor, So we'll just first see what all the ex squares are. So we got an X squared plus X squared. That's two x sward minus two X squared. So that's just zero. So we don't need to worry about the ex Squares anymore. You got two x minus two X That's also zero. So you don't need a way that those anymore. And that just leaves us with the one so we can just simplify this a little bit. One last time we got X Times X plus one. That's our final answer


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