To simplify the expression:
[tex]\[
\frac{6 x^{-2} y^{-3}}{4 x^6 y^{-5}}
\][/tex]
we follow these steps:
1. Separate the coefficients from the variables:
[tex]\[
\frac{6}{4} \cdot \frac{x^{-2}}{x^6} \cdot \frac{y^{-3}}{y^{-5}}
\][/tex]
2. Simplify the coefficients:
[tex]\[
\frac{6}{4} = \frac{3}{2}
\][/tex]
3. Apply the quotient rule for exponents to the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{x^{-2}}{x^6} = x^{-2 - 6} = x^{-8}
\][/tex]
4. Apply the quotient rule for exponents to the [tex]\(y\)[/tex] terms:
[tex]\[
\frac{y^{-3}}{y^{-5}} = y^{-3 - (-5)} = y^{-3 + 5} = y^2
\][/tex]
5. Combine all the simplified parts:
[tex]\[
\frac{3}{2} \cdot x^{-8} \cdot y^2
\][/tex]
6. Express using positive exponents:
[tex]\[
x^{-8} = \frac{1}{x^8}
\][/tex]
So:
[tex]\[
\frac{3}{2} \cdot \frac{1}{x^8} \cdot y^2 = \frac{3y^2}{2x^8}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{\frac{3y^2}{2x^8}}
\][/tex]
