Algebra help!

[BenM]

I am stuck :-(

Could anyone point me in the right direction? This is a review assignment...

Thanks,

Ben

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[Ishuda]

I am stuck :-(
...

This appears to be the problem: Simplify the following
\(\displaystyle \begin{pmatrix}
\frac{5 x^{-2} y^{\frac{2}{3}}}{20 x^{\frac{2}{3}} y^{-\frac{4}{3}}}
\end{pmatrix}^{-\frac{3}{2}}\)

Lets walk through something similar:
\(\displaystyle \begin{pmatrix}
\frac{16 x^{\frac{7}{3}} y^{\frac{1}{3}}}
{\frac{1}{2} x^{-\frac{23}{3}} y^2}
\end{pmatrix}^{-\frac{3}{5}}\)
Let's just work inside the parenthesis first and start by multiply numerator and denominator by 2
\(\displaystyle \begin{pmatrix}
\frac{32 x^{\frac{7}{3}} y^{\frac{1}{3}}}
{x^{-\frac{23}{3}} y^2}
\end{pmatrix}^{-\frac{3}{5}}\)
Now by x^23/3 y^-2
\(\displaystyle \begin{pmatrix}
\frac{32 x^{\frac{7}{3}} y^{\frac{1}{3}} x^{\frac{23}{3}} y^{-2}}
{x^{-\frac{23}{3}} y^2 x^{\frac{23}{3}} y^{-2}}
\end{pmatrix}^{-\frac{3}{5}}\)
= \(\displaystyle \begin{pmatrix}
\frac{32 x^{\frac{7}{3}+\frac{23}{3}} y^{\frac{1}{3}-2} }
{x^{0} y^0}
\end{pmatrix}^{-\frac{3}{5}}\)
= \(\displaystyle (32 x^{\frac{30}{3}} y^{-\frac{5}{3}})^{-\frac{3}{5}}\)
Now take the outside exponent inside
= \(\displaystyle 32^{-\frac{3}{5}} x^{\frac{30}{3} * (-\frac{3}{5})} y^{-\frac{5}{3} * (-\frac{3}{5})}\)
= \(\displaystyle 32^{-\frac{3}{5}} x^{-6} y^{1}\)
= \(\displaystyle (2^5)^{-\frac{3}{5}} x^{-6} y\)
= \(\displaystyle 2^{-3} x^{-6} y\)= \(\displaystyle \frac{y}{2^{3} x^{6}} \)= \(\displaystyle \frac{y}{8 x^6}\)

 

[BenM]

Much thanks!