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I know I have asked a similar question in the past I am stuck on this question.

How would I simplify the following:

$$\left(\frac{xy^{-4}}{2^{-1}x^{-2}}\right)^3\left(\frac{8x^{-2}y^0}{3^{-1}xy^{-3}}\right)^{-2}$$

I have done

$$\frac{x^3y^{-12}}{2^{-3}x^{-6}}\left(\frac{3^{-1}xy^{-3}}{8x^{-2}\cdot 1}\right)^2$$

$$\frac{x^9y^{-12}}{2^{-3}}\frac{3^{-2}x^2y^{-6}}{64x^{-4}}$$

Unfortunately I am not sure how to proceed.

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You can find some good starting points on how to format mathematics on the site here and here. This AMS reference is very useful. If you need to format more advanced things, there are many excellent references on LaTeX on the internet, including StackExchange's own TeX.SE site. –  Zev Chonoles Jan 16 '13 at 2:37
    
@ZevChonoles I wish there was a way to +1 an edit. That was awesome, but also a lot of work. Thanks for clearing up the question! –  anorton Jan 16 '13 at 2:42
    
@anorton: Thanks for your kind words :) I don't want questions to be discriminated against due to poor formatting, and I can write LaTeX pretty quick, so I like to help when I can. –  Zev Chonoles Jan 16 '13 at 2:47

3 Answers 3

up vote 2 down vote accepted

$$\frac{x^9y^{-12}}{2^{-3}}\frac{3^{-2}x^2y^{-6}}{64x^{-4}}$$ Remember that $a^{-n} = \frac{1}{a^n}$. Thus, we have: $$\frac{2^3x^9}{y^{12}}\cdot\frac{x^2x^4}{64\cdot3^2y^6}$$ At this point, we have $a^na^m=a^{n+m}$. I've also changed $64=2^6$. $$\frac{2^3x^{15}}{2^6\cdot3^2y^{18}}$$ Note that we can cancel some of the twos: $$\frac{x^{15}}{2^3\cdot3^2y^{18}}$$ These numbers are easier to work with: $$\frac{x^{15}}{8\cdot9y^{18}}$$ $$\frac{x^{15}}{72y^{18}}$$ Done. :-)

Please let me know if you have any questions. A reminder--please accept the answer that you feel best answers your question (if, in your opinion, one does). This will encourage people to answer your future questions.

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muchas gracias senor. –  Fernando Martinez Jan 17 '13 at 1:43

You're doing well so far! Now you just need to combine the two fractions together, using the rule $$\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$$ Thus, you'll get $$\frac{x^9y^{-12}3^{-2}x^2y^{-6}}{2^{-3}64x^{-4}}$$ I think all the remaining steps after this are ones you've demonstrated knowledge of already, though if you need further help I can add more detail.

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Thanks for help. –  Fernando Martinez Jan 17 '13 at 1:39

Well, the next step would be to get rid of the negative exponents $$\frac{8x^9}{y^{12}}\frac{x^6}{(9)(64)y^6}$$ Now you just simplify by multiplying and reducing the coefficients $$\frac{x^{15}}{72y^{18}}$$ Hope this helps.

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