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REVIEW If $x y^{-2}+y^{-1}=y^{-2},$ then the value of $x$ cannot equal which of the following?

$$

\begin{array}{l}{\mathbf{F}-1} \\ {\mathbf{G}\quad 0} \\ {\mathbf{H}\quad 1} \\ {\mathbf{J} \quad 2}\end{array}

$$

$H$

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Numerade Educator

McMaster University

Baylor University

University of Michigan - Ann Arbor

So this is, Ah pretty normal thing that standardized tests like to pull. They've given us this equation and they're asking us what value X cannot equal but beyond Woods, even asking us knows what they've done here. I don't know about you guys, but it's very rare that I am having to do a whole lot of math with negative exponents because we spend time in algebra class learning about exponents properties, One of which tells us how to take care of negative exponents. Right. Negative exponents really just mean reciprocal. Meaning at least in my class, If I was giving someone a problem like this, the more likely wave that I would write This would be to read it as X over. Why? To the positive second. And then I'd write this as one over why to the positive first and had right that is being equal to one over why? To the positive second, both of these equations that you see here mean exactly the same thing. Neither of them is different. As far as value, they're both equal. But the 2nd 1 is how I think we would. All of us would probably be used to seeing it represented. So, of course, since this is a standardized test, that is a question that is supposed Teoh simulate a standardized test. Of course, they're going to give it to you in a form that already and makes it look tougher than it really is. Okay, so my recommendation would be, Let's look at this, like the second equation with the fractions here. And let's work on it like that. More than likely, you have already done a section both in this glass and maybe an ounce for classes in the past that have taught you how to solve rational equations, meaning equations that have fractions in them. Okay, there's multiple ways to try and do it. Me personally. I try to multiply the entire equation by something that will buy a common denominator. Okay, I try and find a common denominator. Multiply the entire equation by that. So I'm looking here. I have a denominator of y squared. I have a denominator of white of the first and a have denominator of y squared. So the question is what would be a common denominator, meaning what is something that all of those fractions either already have for denominator or that we can multiply up to Right. So in this case, that would be Why squared? Because I already have to denominators of life squared and that other denominator of why we like you just multiply by. Why on top and bottom. And I would get that to denominator of life squared as well. So I'm gonna most play this entire equation by y squared. Or if you prefer, why squared over one? Okay, I'm multiplying every single part of the equation by that. So I'm not actually changing the equation as faras the value is concerned. As long as everything is both played by y squared, it's still okay. So if I take wife square over one times X over y squared well, X over y squared times y squared over one means I have a wife squared in the numerator and I have a wife square in the denominator. Those would cancel each other out, giving me X over one. If I take that for the second term, if I take one over why and multiply that by why squared over one? Well, I have a y squared on top, which is really why times why right. I have just one. Why on the bottom. So one of those wise would cancel out one of them would not leaving me with a why Over one left over from that. Finally, I'm gonna take wife squared over one times. One over y squared one over y squared times y squared over one. I've got a wife squared on top and a wife squared on bottom. Those cancel out, leaving me with just one Now we know X divided by one is X. Why divided by one is why. So we really have X plus y equals one here. Okay, that's what we effectively have. So now let's look at what we're being asked here. It's asking us what value of X can we not have out of those four options. Okay, well, here's the catch. Here. Look at what we started with. Okay? When we wrote this as a fraction, this is what we had right before we started manipulating it. Any of that's where thing Just writing the equation out infraction form in a rational form. That's what we had. Okay, if you guys will remember any time you have a rational equation, there is one thing that we absolutely cannot have happened in our equation where it makes the whole problem undefined. And that would be dividing by zero. Right? Any time you have your dividing by zero, the equation just doesn't work because dividing by zero gives you undefined and it messes up the entire thing. You can't solve it out. Look at what we have for a denominator here. Okay? In this equation, look what we have for Denominator. We've got wise, right? We've got a bunch of wise and our denominator. So the question is, when it's asking us what X can not equal, what we want to do is look at the equation that we have here and say, OK, what value of X would cause why to become zero. Well, if we think of this equation here in terms of why I could just subtract this X over here, and we would have Why equals one minus X. So one minus what gives us zero? Well, we certainly know the answer to that. If ex equaled one, then one minus one would force why Teoh equals zero, right? Right here. One minus one. I would give a zero. So if X is one that Y equals zero. And since the equation they gave us, even though they gave it to us as negative exponents, we all understand that negative exponents really mean we're dealing with fractions, and we can't have zero in the denominator of any part of my fraction. So why can't be zero? Which means if why can't be zero x cannot be won, which is why h would be our answer.

University of Central Missouri