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Fractional Exponents Factor the expression comple…

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Problem 57 Hard Difficulty

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

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Video Transcript

in this problem, we're asked to factor out the lowest power of our variable expression. And so the lowest power. If you compare the 1/2 power and the negative 1/2 power, the lowest power is negative 1/2. So we're going to be factoring out X squared plus one to the negative 1/2. So how do we do that? We factor x squared plus one to the negative 1/2 out of our first term. What will we have left? So how do I factor X squared plus one to the negative. 1/2 out of X squared, plus one to the 1/2 the way I like to think about it is that we would have x squared plus one to the negative 1/2 times X squared, plus one to this something. That's what I'm trying to figure out, which is going to have to equal what we started with X squared, plus one to the 1/2. Now remember that we add exponents in this situation so it would be negative 1/2 plus what gives us positive 1/2. And of course, the answer to that would be one negative 1/2 plus one equals positive 1/2. So that tells us that when we factor out this lowest power of ex term, we're going to be left with X squared plus one to the first so we can substitute that in their X squared plus one to the first. We'll just we don't need to write the first power, but we could and then moving on to the second part, plus two times What? Well, if we're factoring X squared plus one to the negative 1/2 out of here, we're factoring out the whole thing. So all that we have left there is the tube. Okay, Now, we can actually simplify this a little bit, because remember, we don't actually need that one power. They're So these parentheses aren't doing anything for us. The outer ones we do need so we can simplify that inside. Um, factor into X squared. Plus three

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