In Exercises 69–82, simplify each complex rational expression. (1...

Question

In Exercises 69–82, simplify each complex rational expression. (1/(x+h)^2 − 1/x^2)/h

Accepted Answer

Welcome back. I'm so glad you're here. This this is quite the rational expression that we get to simplify. So it starts off with a two X divided by three x minus seven minus X divided by three X plus seven divided by X plus seven divided by nine X squared minus 49. Okay well a couple things we can do. Let's look at that denominator of denominators, the nine X squared minus 49. Let's factor that. So we look at that. That's a difference of squares and we take the square root of each of our squares. So the squared of nine X squared Well that's three X times three X in the first position of our two parentheses and that 49 the square to 49 is going to be a seven and a seven. And just recalling from previous lessons by definition for a difference of squares, one will be plus one will be minus While we factored that next let's work on our numerator. We have two fractions being subtracted. Let's give them a common denominator so that we can do that subtraction. So we're going to take each fraction and multiply it by the other ones denominator. So this first one will be three X plus seven over itself over three X plus seven. And for this second fraction again it's multiplied by the other ones denominator. So three X minus seven over itself. Three X minus seven because when you have something over itself that's just one. It's like we're not even changing this really. Okay so now we can distribute in the numerator. So that will be a two x times three x plus seven over three x plus seven times 3 x -7 minus x times three X minus seven divided by three X plus seven times three X minus seven. Oh my goodness over X plus 7/3 X plus seven times three X minus seven. Oh I can't wait till all these three X plus sevens and minus seven. Hopefully cancel out somewhere. Okay but not yet. We don't want to start canceling out because we're working on the numerator and we want the numerator to combine to one fraction. They have a common denominator now so we can do that. It'll be two x times three X plus seven minus X times three X minus seven. All over our combined numerator denominator which is three X plus seven times three X minus seven. Still over X plus seven over same thing three X plus seven times three X minus seven. Okay let's distribute in our numerator of numerator. So two x times three X. Is six X squared plus two X times seven is 14 X minus x times three X. Is three X squared. And we have a negative X times negative seven which will be a plus seven X. Bring over the rest three x plus seven times 3 x -7 Divided by X-plus seven over three X plus seven times three X minus seven. Okay now let's combine like terms in our numerator of numerator so that six X squared can go with our negative three X squared. Leaving us with three X squared and are plus 14 X plus seven X will give us a plus 21 X bringing over the rest three X plus seven times three X minus seven Divided by X-plus seven over three X plus seven times three X minus seven. And Well looking at our numerator of enumerators we can factor out a 3X. So that'll be three X times Leaving us with X plus seven divided by three X plus seven times three X minus seven divided by X plus seven divided by three X plus seven times three X minus seven. Okay now we have done a lot of simplification here. Now we just have to rethink this a little bit. So we have three x times x plus seven divided by three x plus seven times 3 x - divided by because this big division in the middle we can rewrite that as divided by X plus seven divided by three X plus seven times three X minus seven. And we know that when we are dividing two fractions that's the same as multiplying by the reciprocal of the second one. So the first one is still three X times X plus 7/3 X plus seven times three X minus seven. Now times the reciprocal of the second one. So three X plus seven times three X minus seven over X plus seven. Look at this when they multiply now it is open season on canceling these factors out. If one in the numerator appears in the denominator. Oh yes, here it is. Okay. There's an X plus seven in the numerator and an X plus seven in the denominator. Here is a three X plus seven in the numerator and a three X plus seven in the denominator. Here's a three x minus seven in the numerator and a three x minus seven in the denominator. Those all become fancy shmancy ones now and all we've got left is three X three x times one times one and divided by one. It's three X. I'm going to rewrite that up here so that we can still see it when I scroll up three X. Who would have thunk after all that. Okay. We look at our answer choices and that matches answer choice. A well done. We'll catch you on the next one.

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