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Simplify each complex fraction.$$ rac{x^{-1}+y^{-1}}{(x+y)^{-1}}$$...

Question

Simplify each complex fraction.$$ rac{x^{-1}+y^{-1}}{(x+y)^{-1}}$$

Simplify each complex fraction. $$ \frac{x^{-1}+y^{-1}}{(x+y)^{-1}} $$



Answers

Simplify each complex fraction. $$ \frac{x^{-1}+y^{-1}}{(x+y)^{-1}} $$

So in this questioning of one over one plus X over by which will rewrite is wide plus why over wide the sex over why And then we basically have one over. Why Bust looks over Why and what over something is just gonna be the reciprocal were multiplying by Why times wipe a sex. What we end up with this. Why, over by process starts, we end up with why over wide butt sex as our final solution. But the important thing to know is that we do have a domain restriction in this question, which is? The axe cannot be negative wind. Why cannot be zero. All right, thank you. Expand not be negative. Why and why I cannot be. So I understand why we have these restrictions like let's look at our original problems we have Why over WIPO sex here now in our original question If why was you remember what we were actually given originally was won over one was to worldwide. It's in our original question. If we were going, why zero would have undefined over here? The other thing we're doing is that X is equal If X is equal to negative life for example, of Axis negative. Three Wives Street. We end up with negative ones. Actually, Hawaiian one minus one is zero. So we would have the restriction of one over zero this entire part of her here, one plus x over. Why, that would be serious. Then we still got a domain restriction because we if the denominator zero, it's undefined. So our solution is wild, reckless acts where extra not be negative, why and why cannot be here.

So for this problem, we have to simplify this complex fraction. The first step, but we're going to do is separate. Our new radar and denominator solved each and simplify each separately and then combine them at the end to make our total simplified expression. So first, we're going to simplify numerator. We're going to rewrite it over here, and we can see that we have two different denominators and to subtract them, we need a common denominators. What we're gonna do, we're gonna multiply. Why? Times their first expression X times their second expression. So this will equal to five. Why? Over X y minus two x over x y, which will lead to five wine minus two x for over ex wife. Now we do our denominator again. Most white Why buy the first and exploit a second to make our common denominator now equal to negative for why? Over x y minus six x over x y, which will equal to negative for why minus six x goger x y. Now we can rewrite our fraction to be five. Why? Minus two x over x y divided by negative for why minus six x over x y which means the same thing as five y minus two x over X y times the reciprocal of our bottom two non eater, which is X y over native for Y minus six x this equals two five y minus two x x y and over Native for why minus six x times x Why we see that we have the same term on top and on the bottom of the fractures. We can cancel those, and our final answer is five y minus two X over negative for why minus six X All right, we just rewrite it where the variables air in alphabetical order and it's more simplified. So negative two x plus. Why five? Why over negative too three plus 21 three X plus two Why you can rewrite that and this is your final answer.

So for this problem, we have to simplify this expression. First step is simplifying the numerator and denominator separately. So for the numerator, what we're going to do is first rewrite. And now we see in the subtraction problem, we need a common denominators. What we're going to do is we're going to multiply. Why toe our first expression and extra second expression, giving us a common denominator of X y squared, which equals two. Why minus X over X y squared. Now we go to our denominator. And again we have to find our common denominator. And for these expressions, it's going to be X squared y squared, which means that we're going to multiply. Why toe our first term and x two or second term? This means it's going to be why over x squared y squared minus X over X squared y squared. This will give us why minus X over X squared y squared. Now what we have to do is combined them in our initial expression. This names this will equal to why minus X over X y squared, divided by why minus x over x squared y squared. This expression also equals the following expression which is X minus y over X y squared times the reciprocal of our denominator which is X squared y squared over. Why minus X And now we multiply across. That means this equals two Why minus x times x squared y squared over X y square times why minus x immediately that we see we have the same terms X minus y so you can cancel those out. This equals two x squared. Why squared over X y squared again? We see that we have the same term. Why square to cancel that out And we have expert over. We can cancel out one x from the top and one from the bottom, which means that our final answer is X.

Okay, so for the problem, we want to simplify one over X plus one over why All over X plus y. So we have two fractions in the numerator that we want to get rid of. So we want most fly to cancel them out. So we have X and why I'm a denominator of the fraction. So almost bye bye, ex And why? But we want most by by X and y over text times. Why? So it doesn't change the value. It's just like multiplying by one. So when you do that simple fives one over x times X times Why the X's cancel out and you have why plus 11 over wide times x times Why the wise cancel out and you have X over x times y times X plus y. She's basically saying his wife was fax. So then those cancel out. You're left with one over Thanks times. Why Hunter Answer. Remember, you don't want your denominators to ever equal zero so x plus why cannot equal zero from here and then ex cannot equal zero from there, and why cannot equal here from there


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