SOLUTION: Please help me simplify this rational expression: {{{ ((x^3+y^3)/(2x^3+2x^2y))/((3x^3-3x^2y+3xy^2)/(6x^2-6y^2))}}}

Algebra ->  Algebra  -> Rational-functions -> SOLUTION: Please help me simplify this rational expression: {{{ ((x^3+y^3)/(2x^3+2x^2y))/((3x^3-3x^2y+3xy^2)/(6x^2-6y^2))}}}      Log On

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Question 196861: Please help me simplify this rational expression: +%28%28x%5E3%2By%5E3%29%2F%282x%5E3%2B2x%5E2y%29%29%2F%28%283x%5E3-3x%5E2y%2B3xy%5E2%29%2F%286x%5E2-6y%5E2%29%29
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(21605) About Me  (Show Source):
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
Note: the previous solution has a sign error and the answer should be much simpler.



%28%28x%5E3%2By%5E3%29%2F%282x%5E3%2B2x%5E2y%29%29%2F%28%283x%5E3-3x%5E2y%2B3xy%5E2%29%2F%286x%5E2-6y%5E2%29%29 Start with the given expression.


%28%28x%5E3%2By%5E3%29%2F%282x%5E3%2B2x%5E2y%29%29%28%286x%5E2-6y%5E2%29%2F%283x%5E3-3x%5E2y%2B3xy%5E2%29%29 Multiply the first fraction %28x%5E3%2By%5E3%29%2F%282x%5E3%2B2x%5E2y%29 by the reciprocal of the second fraction %283x%5E3-3x%5E2y%2B3xy%5E2%29%2F%286x%5E2-6y%5E2%29.


%28%28%28x%2By%29%28x%5E2-xy%2By%5E2%29%29%2F%282x%5E3%2B2x%5E2y%29%29%28%286x%5E2-6y%5E2%29%2F%283x%5E3-3x%5E2y%2B3xy%5E2%29%29 Factor x%5E3%2By%5E3 to get %28x%2By%29%28x%5E2-xy%2By%5E2%29.


%28%28%28x%2By%29%28x%5E2-xy%2By%5E2%29%29%2F%282x%5E2%28x%2By%29%29%29%28%286x%5E2-6y%5E2%29%2F%283x%5E3-3x%5E2y%2B3xy%5E2%29%29 Factor 2x%5E3%2B2x%5E2y to get 2x%5E2%28x%2By%29.


%28%28%28x%2By%29%28x%5E2-xy%2By%5E2%29%29%2F%282x%5E2%28x%2By%29%29%29%28%286%28x-y%29%28x%2By%29%29%2F%283x%5E3-3x%5E2y%2B3xy%5E2%29%29 Factor 6x%5E2-6y%5E2 to get 6%28x-y%29%28x%2By%29.


%28%28%28x%2By%29%28x%5E2-xy%2By%5E2%29%29%2F%282x%5E2%28x%2By%29%29%29%28%286%28x-y%29%28x%2By%29%29%2F%283x%28x%5E2-xy%2By%5E2%29%29%29 Factor 3x%5E3-3x%5E2y%2B3xy%5E2 to get 3x%28x%5E2-xy%2By%5E2%29.


%286%28x%2By%29%28x%5E2-xy%2By%5E2%29%28x-y%29%28x%2By%29%29%2F%282x%5E2%2A3x%28x%2By%29%28x%5E2-xy%2By%5E2%29%29 Combine the fractions.


%286%28x%2By%29%28x%5E2-xy%2By%5E2%29%28x-y%29%28x%2By%29%29%2F%286x%5E3%28x%2By%29%28x%5E2-xy%2By%5E2%29%29 Multiply 2x%5E2 and 3x to get 6x%5E3.


%28highlight%286%29%2Ahighlight%28%28x%2By%29%29highlight%28%28x%5E2-xy%2By%5E2%29%29%28x-y%29%28x%2By%29%29%2F%28highlight%286%29x%5E3%2Ahighlight%28%28x%2By%29%29highlight%28%28x%5E2-xy%2By%5E2%29%29%29 Highlight the common terms.


%28cross%286%29%2Across%28%28x%2By%29%29cross%28%28x%5E2-xy%2By%5E2%29%29%28x-y%29%28x%2By%29%29%2F%28cross%286%29x%5E3%2Across%28%28x%2By%29%29cross%28%28x%5E2-xy%2By%5E2%29%29%29 Cancel out the common terms.


%28%28x-y%29%28x%2By%29%29%2F%28x%5E3%29 Simplify


%28x%5E2-y%5E2%29%2F%28x%5E3%29 FOIL



So %28%28x%5E3%2By%5E3%29%2F%282x%5E3%2B2x%5E2y%29%29%2F%28%283x%5E3-3x%5E2y%2B3xy%5E2%29%2F%286x%5E2-6y%5E2%29%29 simplifies to %28x%5E2-y%5E2%29%2F%28x%5E3%29.


In other words, %28%28x%5E3%2By%5E3%29%2F%282x%5E3%2B2x%5E2y%29%29%2F%28%283x%5E3-3x%5E2y%2B3xy%5E2%29%2F%286x%5E2-6y%5E2%29%29=%28x%5E2-y%5E2%29%2F%28x%5E3%29