Simplify each complex fraction. See Examples 5 and 6. x y — + — y...

Question

Simplify each complex fraction. See Examples 5 and 6. x y — + — y x—————— x y — - — y x

Accepted Answer

Hello, everyone. We are asked to simplify the following rational expression. We are given a divided by B minus B divided by A, all of which is divided by A divided by B plus B divided by A. Our answer choices are A, A squared plus two A B plus B squared B A squared minus two A B plus B squared C, the parentheses A squared minus B squared divided by the parentheses A squared plus B squared D, the parentheses A squared plus B squared divided by the parentheses A squared minus B squared. So first, I'm rewriting my original expression. So A divided by B minus B divided by A, divided by A, divided by B plus B divided by A. Next, I'm gonna make a common denominator for my numerator. Um I'm gonna say that that is A B. So I'm gonna have two fractions with the denominator of A B. So in my first fraction to get from just a denominator of B to one of A B I multiplied by A. So I must also multiply my numerator by A. So I'll have a squared divided by A B and for the second fraction in the numerator to get from A to A B I multiplied by B. So I need to multiply my numerator by B which will give me B squared. So my new numerator is A squared divided by A B minus B squared divided by A B. I'm gonna do the same in the denominator A B again will be my common denominator. So to get from B to A B I multiply both top and bottom by A. So I have A squared divided by A B and that's gonna be added to um I'm gonna multiply both top and bottom by B in this case. So B squared divided by A B. Next, I'm gonna write both my numerator and my denominator as a single fraction. So my numerator will be A squared minus B squared divided by A B. My denominator will be A squared plus B squared divided by A B. And now if I multiply both top and bottom by A B, these will cross reduce and cancel out our denominators. So we'll be left with A squared minus B squared divided by A squared plus B squared. And there's no further, we can go with this. So this is our answer A squared minus B squared divided by A squared plus B squared. And that's answer choice C have a nice day.

Simplify each complex fraction. See Examples 5 and 6. x y — + — y x—————— x y — - — y x

Key Concepts

Complex Fractions

Common Denominator

Algebraic Manipulation

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