To solve [tex]\(\left(\frac{10 x^3 y^2}{5 x^{-3} y^4}\right)^{-3}\)[/tex], let's break it down step-by-step.
1. Simplify the inner expression:
[tex]\[
\frac{10 x^3 y^2}{5 x^{-3} y^4}
\][/tex]
Divide the coefficients:
[tex]\[
\frac{10}{5} = 2
\][/tex]
Simplify the exponents of [tex]\(x\)[/tex]:
[tex]\[
\frac{x^3}{x^{-3}} = x^{3 - (-3)} = x^{3 + 3} = x^6
\][/tex]
Simplify the exponents of [tex]\(y\)[/tex]:
[tex]\[
\frac{y^2}{y^4} = y^{2 - 4} = y^{-2}
\][/tex]
Combining all simplified parts, we get:
[tex]\[
2 x^6 y^{-2}
\][/tex]
2. Raise the simplified expression to the power of [tex]\(-3\)[/tex]:
[tex]\[
(2 x^6 y^{-2})^{-3}
\][/tex]
Apply the exponent to each factor:
[tex]\[
2^{-3} \cdot (x^6)^{-3} \cdot (y^{-2})^{-3}
\][/tex]
Simplify each term:
[tex]\[
2^{-3} = \frac{1}{2^3} = \frac{1}{8}
\][/tex]
[tex]\[
(x^6)^{-3} = x^{6 \cdot (-3)} = x^{-18}
\][/tex]
[tex]\[
(y^{-2})^{-3} = y^{-2 \cdot (-3)} = y^6
\][/tex]
3. Combine the simplified terms:
[tex]\[
\frac{1}{8} \cdot x^{-18} \cdot y^6
\][/tex]
Rewrite the expression:
[tex]\[
\frac{y^6}{8 x^{18}}
\][/tex]
Therefore, the simplified and final form of the given expression is:
[tex]\[
\frac{y^6}{8 x^{18}}
\][/tex]
So, the correct answer is:
[tex]\(\boxed{\frac{y^6}{8 x^{18}}}\)[/tex]
