To determine which expression is equivalent to [tex]\(\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}\)[/tex], we'll break the problem into simplifying the coefficients and simplifying the powers of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
1. Simplify the Coefficients:
[tex]\[
\frac{-18}{-12}
\][/tex]
Since both the numerator and the denominator are negative, dividing them will yield a positive result. Thus:
[tex]\[
\frac{-18}{-12} = \frac{18}{12} = \frac{3}{2}
\][/tex]
2. Simplify the Powers of [tex]\(a\)[/tex]:
When dividing like bases, we subtract the exponents:
[tex]\[
\frac{a^{-2}}{a^{-4}} = a^{-2 - (-4)} = a^{-2 + 4} = a^2
\][/tex]
3. Simplify the Powers of [tex]\(b\)[/tex]:
Similarly, for [tex]\(b\)[/tex]:
[tex]\[
\frac{b^5}{b^{-6}} = b^{5 - (-6)} = b^{5 + 6} = b^{11}
\][/tex]
Now, combining these simplifications we get:
[tex]\[
\frac{3}{2} \cdot a^2 \cdot b^{11}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{3 a^2 b^{11}}{2}
\][/tex]
Therefore, the expression equivalent to [tex]\(\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}\)[/tex] is:
[tex]\[
\boxed{\frac{3 a^2 b^{11}}{2}}
\][/tex]
