Simplifying a Rational Expression

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humanbean

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SOLVED: Simplifying a Rational Expression

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Hello! Here's a problem I've tried to solve about three or four times now. I need to simplify this expression:

((1/a^2)+(2/ab)+(1/b^2))/((1/a^2)-(1/b^2))

I've been trying to solve it by using regular order of operation, and by getting their denominators the same. But I keep ending up with big, unwieldy terms that I can't do much with. The correct answer, according to the book, is:

(b+a)/(b-a)
 
Last edited:
humanbean said:
Hello! Here's a problem I've tried to solve about three or four times now. I need to simplify this expression:

((1/a^2)+(2/ab)+(1/b^2))/((1/a^2)-(1/b^2))

I've been trying to solve it by using regular order of operation, and by getting their denominators the same. But I keep ending up with big, unwieldy terms that I can't do much with. The correct answer, according to the book, is:

(b+a)/(b-a)
Click to expand...

Hint:

(x + y)2 = x2 + y2 + 2xy

(x + y)(x - y) = x2 - y2
 
humanbean said:
Hello! Here's a problem I've tried to solve about three or four times now. I need to simplify this expression:

((1/a^2)+(2/ab)+(1/b^2))/((1/a^2)-(1/b^2))

I've been trying to solve it by using regular order of operation, and by getting their denominators the same. But I keep ending up with big, unwieldy terms that I can't do much with. The correct answer, according to the book, is:

(b+a)/(b-a)
Click to expand...
Look at this webpage.

Do you realize that the expression reduces to
\(\displaystyle \dfrac{{\left( {\frac{1}{a} + \frac{1}{b}} \right)^2 }}{{\left( {\frac{1}{a} + \frac{1}{b}} \right)\left( {\frac{1}{a} - \frac{1}{b}} \right)}} = \dfrac{{\left( {\frac{1}{a} + \frac{1}{b}} \right)}}{{\left( {\frac{1}{a} - \frac{1}{b}} \right)}} = \dfrac{{b + a}}{{b - a}}
\)
 
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