SOLUTION: 1-(1+xy^-2)^-2 divided by 1+(1+xy^-2)^-2 Directions are to simplify. Thanks

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Question 144240: 1-(1+xy^-2)^-2 divided by 1+(1+xy^-2)^-2
Directions are to simplify. Thanks
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!



1-%281%2Bxy%5E%28-2%29%29%5E%28-2%29÷1%2B%281%2Bxy%5E%28-2%29%29%5E%28-2%29 



Write both terms xy%5E%28-2%29 as x%2Fy%5E2



1-%281%2Bx%2Fy%5E2%29%5E%28-2%29÷1%2B%281%2Bx%2Fy%5E2%29%5E%28-2%29



Write the 1's inside the parentheses as 1%2F1



1-%281%2F1%2Bx%2Fy%5E2%29%5E%28-2%29÷+1%2B%281%2F1%2Bx%2Fy%5E2%29%5E%28-2%29



Get an LCD of y%5E2 inside the parentheses and

multiply top and bottom of both 1%2F1's by y%5E2



+1-%28%281%2Ay%5E2%29%2F%281%2Ay%5E2%29%2Bx%2Fy%5E2%29%5E%28-2%29÷+1%2B%28%281%2Ay%5E2%29%2F%281%2Ay%5E2%29%2Bx%2Fy%5E2%29%5E%28-2%29



1-%28y%5E2%2Fy%5E2%2Bx%2Fy%5E2%29%5E%28-2%29÷+1%2B%28%28y%5E2%29%2F%28y%5E2%29%2Bx%2Fy%5E2%29%5E%28-2%29



Combine the numerators over the LCD in the parentheses:



+1-%28%28y%5E2%2Bx%29%2Fy%5E2+%29%5E%28-2%29÷+1%2B%28%28y%5E2%2Bx%29%2Fy%5E2%29%5E%28-2%29



Now use this principle on those fractions to negative powers:

                %28A%2FB%29%5E%28-N%29+=+%28B%2FA%29%5EN



+1-%28y%5E2%2F%28y%5E2%2Bx%29+%29%5E2÷+1%2B%28y%5E2%2F%28y%5E2%2Bx%29%29%5E2



Square tops and bottoms:



+1-y%5E4%2F%28y%5E2%2Bx%29%5E2÷+1%2By%5E4%2F%28y%5E2%2Bx%29%5E2



Write the 1's as 1%2F1



+1%2F1-y%5E4%2F%28y%5E2%2Bx%29%5E2÷+1%2F1%2By%5E4%2F%28y%5E2%2Bx%29%5E2



Get an LCD of %28y%5E2%2Bx%29%5E2 and multiply the 1%2F1's

top and bottom by it:



+%281%2A%28y%5E2%2Bx%29%5E2%29%2F%281%2A%28y%5E2%2Bx%29%5E2%29-y%5E4%2F%28y%5E2%2Bx%29%5E2÷%281%2A%28y%5E2%2Bx%29%5E2%29%2F%281%2A%28y%5E2%2Bx%29%5E2%29%2By%5E4%2F%28y%5E2%2Bx%29%5E2



+%28y%5E2%2Bx%29%5E2%2F%28y%5E2%2Bx%29%5E2-y%5E4%2F%28y%5E2%2Bx%29%5E2÷%28y%5E2%2Bx%29%5E2%2F%28y%5E2%2Bx%29%5E2%2By%5E4%2F%28y%5E2%2Bx%29%5E2



Combine the numerators over the LCD



+%28%28y%5E2%2Bx%29%5E2-y%5E4%29%2F%28y%5E2%2Bx%29%5E2÷%28%28y%5E2%2Bx%29%5E2%2By%5E4%29%2F%28y%5E2%2Bx%29%5E2



Invert the second fraction and change division to multiplication:



+%28%28y%5E2%2Bx%29%5E2-y%5E4%29%2F%28y%5E2%2Bx%29%5E2×%28y%5E2%2Bx%29%5E2%2F%28%28y%5E2%2Bx%29%5E2%2By%5E4%29



Now the %28y%5E2%2Bx%29%5E2's cancel



+%28%28y%5E2%2Bx%29%5E2-y%5E4%29%2Fcross%28%28y%5E2%2Bx%29%5E2%29×cross%28%28y%5E2%2Bx%29%5E2%29%2F%28%28y%5E2%2Bx%29%5E2%2By%5E4%29



+%28%28y%5E2%2Bx%29%5E2-y%5E4%29%2F%28%28y%5E2%2Bx%29%5E2%2By%5E4%29 



Now we square +%28y%5E2%2Bx%29%5E2+ = +%28y%5E2%2Bx%29%28y%5E2%2Bx%29+ = +y%5E4%2B2xy%5E2%2Bx%5E2+



And replace in the numerator and denominator



+%28%28y%5E4%2B2xy%5E2%2Bx%5E2%29-y%5E4%29%2F%28%28y%5E4%2B2xy%5E2%2Bx%5E2%29%2By%5E4%29



+%28y%5E4%2B2xy%5E2%2Bx%5E2-y%5E4%29%2F%28y%5E4%2B2xy%5E2%2Bx%5E2%2By%5E4%29



Combine terms in numerator and denominator



+%282xy%5E2%2Bx%5E2%29%2F%282y%5E4%2B2xy%5E2%2Bx%5E2%29



Edwin