Perform all indicated operations, and write each answer with posi...

Question

Perform all indicated operations, and write each answer with positive integer exponents. [ (x^-2 + y^-2)/ (x^-2 - y^-2) ] * [ (x+y)/(x-y) ]

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to simplify the expression A. To the negative two minus B to the negative two divided by A. To the negative two plus B to the negative two times A plus B divided by a minus B. So in this case you can see that there's two sort of components to this expression, we have these two fractions. So in order to make this simplification easier, I'm first just going to take a look at this first fraction which is A to the negative two minus B to the negative two divided by A. to the -2 Plus B to the -2. So from here you can see that this first fraction has variables with negative exponents. So I'm going to want to get rid of those negative exponents. So recall that if a variable let's say X has a negative exponent say negative end. I can write this as a fraction with X. To the negative end becomes one divided by X. To the end power and notice how when I put it in the denominator of the fraction the end is now positive. So I'm gonna want to do that for my expression. So a to the -2 becomes one divided by a squared with a positive too because it's now in the denominator minus one divided by B. To the negative two is now one divided by B squared once again positive. And my denominator is one divided by a squared again plus one divided by B squirt. So I wrote these in different colors. Just because it seems to be getting a little complicated. So go ahead and solve for or simplify my numerator and my denominator separately for a little bit. Just to make sure that I simplify correctly. So you can see that my numerator, I have these two fractions. And in order to simplify them, I want to try and combine them well. In order to combine fractions, I need to make sure that they have a common denominator. So that means that I need to multiply this first fraction with a B squared and the second fraction with an A squared. So if I do this multiplication, this first fraction becomes B squared divided by a squared times b squared -2 fraction, which is a squared divided by a squared times B squared. And I can see that they have a common denominator, so I can combine them and if I combine them this becomes b squared minus a squared, divided by a squared times B squared. And I can't simplify that any further. So let's go back and look at the denominator and you can see the denominator has the same issue where we have two fractions that don't have the same denominator. So I need to multiply this first fraction by B squared over b squared. And the second fraction. Buy a sward over a squared. And if I do that multiplication. The denominator becomes b squared divided by a squared times B squared plus a squared divided by a squared times B squared. And now they have the same denominator so I can combine them. So this becomes B squared plus a squared, divided by a squared times b squared. Well now I have these two fractions on top of each other. So if I rewrite them in a more linear format, so b squared minus a squared, divided by a squared times B squared. And I say divided by the second fraction, which is B squared plus a squared divided by a squared times B squared. Recall that I can change this division to multiplication as long as I switch the numerator and denominator of the second fraction. So the second fraction is now going to be a squared times B squared divided by b squared plus a squared. And from here you can see that a squared times b squared actually cancels out. So we're left with just b squared minus a squared, divided by b squared plus a squared. And this numerator here should look a little familiar because it's actually a difference of squares. So recall that if you have a difference of squares, say x squared minus y squared, you can rewrite this as X plus y, times x minus y. So we can rewrite this numerator as b minus a times B plus A and this is still over B squared plus a squared. But recall that this isn't the only portion of our expression we still need the second half here. So let's put it back in. So if we do that, let's take the first part which is the b minus A. Times B plus A divided by B squared plus a squared times the second portion which is A plus B, divided by a minus B. Well let's see if we can simplify any of these expressions out and none of them, we can't get rid of a minus B from this numerator because a -7 is not there. But if we factor out a negative from this expression this becomes negative B plus A. So if I rewrite the A. First would be a minus B and that looks very similar. Well that is a minus B down here. So if I factor out this negative, I can get rid of the A minus B. So I'll go ahead and do that. So now these cancel out and I'm left with this negative times B plus a times A plus B, divided by B squared plus a squared. And if we look at B plus A and A plus B, I can rewrite this as a plus B. So in reality this numerator is negative, A plus B squared divided by B squared plus a squared. So finally if we look at this expression we can't simplify any further. So this becomes our final answer for simplifying our expression and you can see that that aligns with answer choice. Deep. So hopefully this video helped and see you next time.

Perform all indicated operations, and write each answer with positive integer exponents. [ (x^-2 + y^-2)/ (x^-2 - y^-2) ] * [ (x+y)/(x-y) ]

Key Concepts

Negative Exponents

Factoring Differences of Squares

Simplifying Rational Expressions

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