To find the equivalent expression for [tex]\(\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}\)[/tex], we need to simplify the fraction carefully, following algebraic rules and properties of exponents.
1. Simplify the constants:
[tex]\[
\frac{-18}{-12}
\][/tex]
The negative signs cancel out, simplifying to:
[tex]\[
\frac{18}{12}
\][/tex]
Further simplification of the fraction gives:
[tex]\[
\frac{18 \div 6}{12 \div 6} = \frac{3}{2}
\][/tex]
2. Simplify the exponents for [tex]\(a\)[/tex]:
[tex]\[
\frac{a^{-2}}{a^{-4}}
\][/tex]
Using the properties of exponents, specifically [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we get:
[tex]\[
a^{-2 - (-4)} = a^{-2 + 4} = a^2
\][/tex]
3. Simplify the exponents for [tex]\(b\)[/tex]:
[tex]\[
\frac{b^5}{b^{-6}}
\][/tex]
Using the same exponent rule:
[tex]\[
b^{5 - (-6)} = b^{5 + 6} = b^{11}
\][/tex]
4. Combine the simplified components:
Putting it all together, we have:
[tex]\[
\frac{3}{2} \cdot a^2 \cdot b^{11}
\][/tex]
Thus, the expression [tex]\(\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}\)[/tex] simplifies to:
[tex]\[
\frac{3 a^2 b^{11}}{2}
\][/tex]
The correct choice from the given options is:
[tex]\[
\boxed{\frac{3 a^2 b^{11}}{2}}
\][/tex]
