Sure, let's simplify the given expression step-by-step. We need to simplify the expression:
[tex]\[
\frac{\left(3 x y^{-2}\right)^2}{2 x^{-3} y^3}
\][/tex]
1. Simplify the numerator:
[tex]\[
\left(3 x y^{-2}\right)^2
\][/tex]
When we square a term, we square each factor inside the parentheses:
[tex]\[
(3 x y^{-2})^2 = 3^2 \cdot x^2 \cdot (y^{-2})^2 = 9 \cdot x^2 \cdot y^{-4}
\][/tex]
So the numerator simplifies to:
[tex]\[
9 x^2 y^{-4}
\][/tex]
2. Simplify the denominator:
[tex]\[
2 x^{-3} y^3
\][/tex]
This expression is already simplified.
3. Combine the results:
We need to divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{9 x^2 y^{-4}}{2 x^{-3} y^3}
\][/tex]
4. Simplify the division:
The division of the terms involves subtracting the exponents for each corresponding variable:
[tex]\[
\frac{9 x^2}{2 x^{-3}} \cdot \frac{y^{-4}}{y^3} = \frac{9 x^{2 - (-3)}}{2} \cdot y^{-4 - 3}
\][/tex]
Simplifying the exponents, we get:
[tex]\[
\frac{9 x^{2 + 3}}{2} \cdot y^{-7} = \frac{9 x^5}{2} \cdot y^{-7}
\][/tex]
5. Rewriting with positive exponents:
Negative exponents can be rewritten as positive exponents by moving the term to the denominator:
[tex]\[
\frac{9 x^5}{2 y^7}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{\frac{9 x^5}{2 y^7}}
\][/tex]
