Multiply or divide, as indicated. (x^2 - y^2)/(x - y)^2 * (x^2 - ...

Question

Multiply or divide, as indicated. (x^2 - y^2)/(x - y)^2 * (x^2 - xy + y^2)/(x^2 - 2xy + y^2) ÷ (x^3 + y^3)/(x - y)^4

Accepted Answer

Hey everyone here we are asked to simplify the given expression by applying the operations indicated. So here we are given the expression where we have a squared minus nine B squared all divided by the quantity of a minus three B squared times another fraction where we have the numerator of a squared minus three A B plus nine B squared, all divided by a squared minus six A B plus nine B squared. And this is divided by our third fraction with the numerator of a cubed plus 27 B cubed, all divided by the quantity of a minus three B raised to the fourth power. Now our first step here before we begin simplifying, we need to recall that when we divide by a fraction, it is the same as multiplying by the reciprocal. So what we need to do here is multiply by the reciprocal of our third fraction. So rewriting our expression, we again had a squared minus nine B squared, all divided by the quantity of of a minus three B squared times the quantity of a squared minus three A B plus nine B squared, All divided by the quantity of a squared minus six A B plus nine B squared. And now again we need to multiply by the reciprocal of our third fraction. So the numerator becomes the quantity of a minus three B raised to the fourth power. And it is all divided by the quantity of a cubed plus 27 B cubed. So now the best way to go about simplifying this problem is to search for cancel a ble terms. So the first step here is we want to expand the first numerator and here we can see that we actually have the difference of two squares. We have a squared minus nine B squared. So to expand this expression we need to recall the difference of squares formula which tells us that a squared minus B squared is equivalent to the quantity of a plus B times the quantity of a minus B. So now with this formula in mind we can rewrite our first numerator. So again according to the formula, our first numerator becomes the quantity of a minus three B times the quantity of a plus three B. And this is all divided by the same denominator where we have the quantity of a minus three B squared. And now this is multiplied by our second fraction here where we had the numerator as a squared minus three A B plus nine B squared. And this is all divided by a squared minus six A B plus nine B squared. And again we need to multiply our third fraction here where we had the numerator as the quantity of a minus three B raised to the fourth power. All div divided by a cute plus 27. Be cute. So now looking at what we have so far we can see that if we were to group together this quantity of a plus three B from our first numerator with the entire second numerator. We see that we actually have the factored form of the sum of two cubes. So we need to recall the sum of two cubes formula which tells us that X cubed plus y cubed is equivalent to the quantity of X plus y times the quantity of x squared minus X, Y plus y squared. And so now we just need to compare this formula with our expression and we see that X is going to be A in our case and then Y is going to be our three B. And so now we see our expression where we had the quantity of a plus three B times the quantity of a squared minus three A B plus nine B squared. We see that according to our formula this expression reduces to a cute Plus the quantity of three B cubed. And this simplifies to just a cubed plus b cubed. So now with this reduced form we can go ahead and use it to rewrite our expression. So our expression becomes the quantity of a minus three B divided by the quantity of a minus three B squared times our second fraction here which now has a numerator of a cubed plus 27. Be cute. And it is all divided by the same denominator where we had a squared minus six A B plus nine B squared. And we multiply by our third fraction which remains as the quantity of a minus three B raised to the fourth power. All divided by a cubed plus 27 B cubed. And so now from here we can start making some cancelations. So we see that the reduced form, our reduced second numerator now cancels with the denominator of our third fraction here. So we have the a cubed plus 27 B cubed. Both cancel. And we also see that in the denominator of our first fraction we have the quantity of a minor three B squared. And this reduces the numerator of our third fraction. But since the quantity of a minus three B in the numerator of the third fraction is raised to the fourth power. It just reduces to being raised to the second power. So now we can rewrite our expression with the terms we have left. So we have the quantity of a minus three B times our second fraction which becomes one divided by a squared minus six A B plus nine B squared. And now our third fraction becomes times just this quantity of a minus three B squared. So now from here we want to expand our third term here we want to expand this quantity of a minus three B squared. And so doing this, our expression becomes the quantity of a. The quantity of a minus three B times one divided by a squared minus six A B plus nine B squared times. And now our expanded form is the quantity of a squared minus six A. B plus nine B squared. And with this we C we can cancel the numerator or the denominator of our second fraction with our third term here or third expression here. So doing this our final expression, We are just left with this a - b. And with this we have fully simplified our original expression and we are left here with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Multiply or divide, as indicated. (x^2 - y^2)/(x - y)^2 * (x^2 - xy + y^2)/(x^2 - 2xy + y^2) ÷ (x^3 + y^3)/(x - y)^4

Key Concepts

Factoring Polynomials

Rational Expressions

Properties of Exponents

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