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## smurfy14 5 years ago [(1/(x^2)-1/Y^2)]/[(1/x^2+2/xy+1/y^2)] plz help! youd be awsome if ya did! (tell me how you get then answer as well, thanks!)

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1. anonymous

What are your instructions for this problem?

2. smurfy14

"simplify complex functions"

3. anonymous

OK, I started by multiplying the top and bottom of the complex fraction by the LCD, which is x^2y^2 in this case

4. anonymous

$\frac{\frac{1}{x^2}-\frac{1}{2}}{\frac{1}{x^2}+\frac{2}{xy}+\frac{1}{y^2}}$

5. anonymous

arya has it, just wanted to type it.

6. anonymous

Which came out to (y^2-x^2)/(y^2+2xy+x^2)

7. anonymous

Sorry, haven't figured out how to type all of the nice equations yet

8. anonymous

there is a gimmick here. the denominator is $(\frac{1}{x}+\frac{1}{y})^2$

9. anonymous

and the numerator is $(\frac{1}{x}+\frac{1}{y})(\frac{1}{x}-\frac{1}{y})$

10. anonymous

so you can cancel.

11. smurfy14

can you explain how multiplying the LCD with the second fraction on the bottom came out to be 2xy?

12. smurfy14

nevermind i got it

13. anonymous

$(2/xy)timesx ^{2}y ^{2}=2x ^{2}y ^{2}/xy=2$

14. smurfy14

idk what to do after (y^2-x^2)/(y^2+2xy+x^2)

15. anonymous

Ignore my response, it didn't come out quite right

16. anonymous

Factor

17. anonymous

$\frac{a^2-b^2}{a+2ab+b^2}=\frac{(a+b)(a-b)}{(a+b)(a+b)}=\frac{a-b}{a+b}$

18. anonymous

y^2-x^2=(y-x)(y+x) y^2+2xy+x^2=(y+x)(y+x)

19. anonymous

$\frac{\frac{1}{x}-\frac{1}{y}}{\frac{1}{x}+\frac{1}{y}}$ multiply top an bottom by $xy$

20. anonymous

Then we can cancel one of the (y+x)

21. smurfy14

oh ok thanks so much! think you could help me with [a-b]/[a^-1=b^-1]

22. anonymous

Sure

23. smurfy14

imean [a-b]/[a^-1-b^-1]

24. anonymous

OK, a^-1=1/a and b^-1=1/b

25. anonymous

Does that part make sense?

26. smurfy14

no? so ur saying you just move the -1 to the top and not the a along with it?

27. anonymous

Not quite, lets see if it makes more sense when I type out the whole equation. Hang on

28. anonymous

(a-b)/(a^-1-b^-1) = (a-b)/[(1/a)-(1/b)]

29. anonymous

Does that make any more sense or no?

30. smurfy14

oh ok ya that makes sense, so you dont move the a with the 1 im guessing

31. anonymous

$\frac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{x y}}=1-\frac{2 x}{x+y}$ Both equation sides evaluated at x=11 and y=17 yield the value 3/14. Note: The first "y" from left to right in the problem expression text string is a cap y, Y, not lower case y. Mathematica views the two as different characters. The cap Y was changed to lower case prior to solving the problem.

32. anonymous

Not quite sure what you mean

33. smurfy14

hah nevermind i got it you can keep going :)

34. anonymous

Cool, so then just like the previous problem the next step is to multiply the to and bottom by the LCD, which in this case is ab

35. anonymous

*top

36. smurfy14

so (ab-ab)/(0)?

37. smurfy14

how do you figure out the denominator?

38. anonymous

I got (a-b)ab for the numerator and b-a for the denominator

39. anonymous

To get the denominator: (1/a-1/b)ab (the LCD)=ab/a-ab/b=b-a

40. smurfy14

ok got it so how did you get the numerator?

41. anonymous

I just multiplied (a-b) by the LCD: ab and got (a-b)ab

42. smurfy14

oh duh lol so would you simplify that farther?

43. anonymous

Yes, at this point the equation looks like [(a-b)ab]/(b-a) and if we factor out -1 from the denominator we get [(a-b)ab]/[-(a-b)]

44. anonymous

Then the (a-b) cancels and we get -ab

45. smurfy14

oh ok thank you soo much!!

46. anonymous

No problem, glad I could help! :)

47. anonymous

$\frac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{x y}}=\frac{\frac{-x^2+y^2}{x^2 y^2}}{\frac{x^2+2 x y+y^2}{x^2 y^2}}=\frac{-x^2+y^2}{x^2+2 x y+y^2}=\frac{(-(x-y) )}{(x+y)}=\frac{-x+y}{x+y}$$1-\frac{2 x}{x+y}=\frac{-x+y}{x+y}$

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