Simplify each complex fraction. [ 1/(x^3-y^3) ] / [ 1/(x^2 -y^2) ...

Question

Simplify each complex fraction. [ 1/(x^3-y^3) ] / [ 1/(x^2 -y^2) ]

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to simplify this expression. That is a big fraction where we have eight divided by a squared minus B squared in the numerator and eight divided by A. To the third plus B to the third in the denominator. So because we have fractions in both the numerator and the denominator of this expression, I'm gonna separate them to make it easier to simplify. So if I look at the numerator I have eight divided by a squared minus B squared. And if you think about this form a squared minus B squared, it's very similar to the difference of squares which follows the form X squared minus Y squared. And the difference of squares can also be written as X plus Y times X minus. Why? So that means if we think about a squared minus B squared, we can also write it as a plus B Times A. -7. So that means I'll rewrite my numerator as eight divided by A plus B Times A. -7. And I can't really do much else with my numerator. So let's take a look at the denominator which is eight divided by eight to the third plus B to the third. And while we had the difference of squares for the numerator we have the denominator which follows more closely the sum of cubes. So the sum of cubes has the general form X to the third plus Y to the third. And we can rewrite that as X plus Y times X squared minus X, Y plus y squared. So using the sum of cubes, we can also rewrite A to the third plus B to the third. And we can rewrite it as A plus B times a squared minus a B plus b squared. So that means that my denominator will be eight divided by a plus B times a squared minus a B plus B squared. And I can't do much else with my denominator, so I'll go ahead and recombine my numerator and my denominator. So my numerator was eight divided by a plus B times a minus B. And I'll go ahead and write this in more of a Linear format for division instead of in the fraction form. So that's being divided by my denominator, which is eight divided by a plus B times a squared minus a B plus b squared. And recall that when we have this division in the linear format and we have fractions, I can change this to a multiplication. If I switch the Numerator and denominator of my 2nd fraction. So the second fraction will now be a plus B times a squared minus a B plus b squared, divided by eight. And with this new format you can see that I have an eight in the numerator of my first fraction. And in the denominator of my second fraction, which means that I can cancel those eights out and then I also have an A plus B. In the denominator of my first fraction, and in the numerator of my second fraction, so I can also cancel out A plus B. So that leaves me with a squared minus a B plus b squared Divided by A. And you can't simplify this any further. So this becomes my final simplification. And if we look at the answer choices, you can see that answer choice. D also has a squared minus a B plus B squared divided by a minus B. So hopefully this video helped and see you next time.

Simplify each complex fraction. [ 1/(x^3-y^3) ] / [ 1/(x^2 -y^2) ]

Key Concepts

Complex Fractions

Difference of Squares

Factoring Polynomials

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