# Simplify algebra exponent

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! – apnorton 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

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

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.
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.