To simplify the given expression:
[tex]\[
\left(\frac{\left(-x^2 y^3\right)^{-2}\left(2 x y^5\right)^2}{\left(-3 x^3 y^4\right)^4\left(2 x^6 y^{-2}\right)^{-1}}\right)^{-3}
\][/tex]
let's break it down step-by-step.
1. Simplify the numerator [tex]\(\left(-x^2 y^3\right)^{-2}\left(2 x y^5\right)^2\)[/tex]:
[tex]\[
\left(-x^2 y^3\right)^{-2} = \left(-1 \cdot x^2 \cdot y^3\right)^{-2} = (-1)^{-2} \cdot (x^2)^{-2} \cdot (y^3)^{-2} = 1 \cdot x^{-4} \cdot y^{-6} = x^{-4} y^{-6}
\][/tex]
[tex]\[
\left(2 x y^5\right)^2 = 2^2 \cdot (x)^2 \cdot (y^5)^2 = 4 \cdot x^2 \cdot y^{10} = 4x^2 y^{10}
\][/tex]
Now, multiply these results together:
[tex]\[
x^{-4} y^{-6} \cdot 4 x^2 y^{10} = 4 \cdot x^{-4+2} \cdot y^{-6+10} = 4 \cdot x^{-2} \cdot y^4
\][/tex]
2. Simplify the denominator [tex]\(\left(-3 x^3 y^4\right)^4 \left(2 x^6 y^{-2}\right)^{-1}\)[/tex]:
[tex]\[
\left(-3 x^3 y^4\right)^4 = (-3)^4 \cdot (x^3)^4 \cdot (y^4)^4 = 81 \cdot x^{12} \cdot y^{16} = 81x^{12} y^{16}
\][/tex]
[tex]\[
\left(2 x^6 y^{-2}\right)^{-1} = \frac{1}{2 x^6 y^{-2}} = \frac{1}{2} \cdot x^{-6} \cdot y^2 = \frac{1}{2} x^{-6} y^2
\][/tex]
Now, multiply these results together:
[tex]\[
81 x^{12} y^{16} \cdot \frac{1}{2} x^{-6} y^2 = 81 \cdot \frac{1}{2} \cdot x^{12-6} \cdot y^{16+2} = \frac{81}{2} \cdot x^6 \cdot y^{18} = \frac{81}{2} x^6 y^{18}
\][/tex]
3. Rewrite the entire fraction and simplify:
[tex]\[
\frac{4x^{-2} y^4}{\frac{81}{2} x^6 y^{18}} = \frac{4x^{-2} y^4 \cdot 2}{81 x^6 y^{18}} = \frac{8 x^{-2} y^4}{81 x^6 y^{18}} = \frac{8}{81} \cdot \frac{x^{-2}}{x^6} \cdot \frac{y^4}{y^{18}}
\][/tex]
Simplify further:
[tex]\[
= \frac{8}{81} \cdot x^{-2-6} \cdot y^{4-18} = \frac{8}{81} x^{-8} y^{-14}
\][/tex]
4. Apply the outer exponent [tex]\(-3\)[/tex]:
[tex]\[
\left(\frac{8}{81} x^{-8} y^{-14}\right)^{-3} = \left(\frac{81}{8} \cdot x^8 y^{14}\right)^3 = \left(\frac{81}{8}\right)^3 \cdot (x^8)^3 \cdot (y^{14})^3 = \frac{81^3}{8^3} \cdot x^{24} y^{42}
\][/tex]
Calculate [tex]\(81^3\)[/tex] (note: [tex]\(81 = 3^4\)[/tex], so [tex]\(81^3 = (3^4)^3 = 3^{12}\)[/tex]):
[tex]\[
81^3 = 531441
\][/tex]
Calculate [tex]\(8^3\)[/tex]:
[tex]\[
8^3 = 512
\][/tex]
Putting it all together:
[tex]\[
\left(\frac{81}{8}\right)^3 x^{24} y^{42} = \frac{531441}{512} x^{24} y^{42}
\][/tex]
Thus, the final simplified expression is:
[tex]\[
\boxed{\frac{531441}{512} x^{24} y^{42}}
\][/tex]
