$$\frac{36 x^2-81 y^2}{216 x^3-729 y^3} $$

Answer will be in the form $\frac{A}{B}$ and the answer should be shortened as much as possible

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    $\begingroup$ $36 x^2-81 y^2 = (6x)^2-(9y)^2$, $216 x^3-729 y^3 = (6x)^3-(9y)^3$. $\endgroup$ – njguliyev Sep 30 '13 at 9:36
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    $\begingroup$ That's a good hint, @njguliyev. Make it an answer? $\endgroup$ – Cameron Buie Sep 30 '13 at 9:46

Okay, I suppose I will, since njguliyev won't (and since I planned to give that very hint, initially).

The big hint here is that: $$36x^2-81y^2=(6x)^2-(9y)^2\\216x^3-729y^3=(6x)^3-(9y)^3$$

What do you know about how to factor a difference of squares? a difference of cubes?

  • $\begingroup$ That's the problem, today my brain doesn't work. $\endgroup$ – magnus Sep 30 '13 at 11:28
  • $\begingroup$ Well, $a^2-b^2=(a-b)(a+b)$. Have you ever seen that before? You've likely seen the way to factor $a^3-b^3,$ too. $\endgroup$ – Cameron Buie Sep 30 '13 at 11:32
  • $\begingroup$ I think it is (a-b)(a+b)(a+b), but i don't get $\endgroup$ – magnus Sep 30 '13 at 11:41
  • $\begingroup$ Not quite. Rather, it is $a^3-b^3=(a-b)(a^2+ab+b^2).$ $\endgroup$ – Cameron Buie Sep 30 '13 at 15:32
  • $\begingroup$ Ok, but I don't understand what the next step is and how to solve the problem. Would be extremely relieved if you could help me. I don't understand why I don't get it $\endgroup$ – magnus Sep 30 '13 at 18:05

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