To simplify the given expression [tex]\(\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}\)[/tex], we will proceed step by step.
1. Simplify the coefficients:
The coefficients are [tex]\(-18\)[/tex] in the numerator and [tex]\(-12\)[/tex] in the denominator.
[tex]\[
\frac{-18}{-12} = \frac{18}{12} = \frac{3}{2}
\][/tex]
The simplified coefficient is therefore [tex]\(\frac{3}{2}\)[/tex].
2. Simplify the variable [tex]\(a\)[/tex]:
The numerator has [tex]\(a^{-2}\)[/tex] and the denominator has [tex]\(a^{-4}\)[/tex].
Using the properties of exponents, namely [tex]\(a^m / a^n = a^{m-n}\)[/tex],
[tex]\[
\frac{a^{-2}}{a^{-4}} = a^{-2 - (-4)} = a^{-2 + 4} = a^{2}
\][/tex]
So, the simplified expression involving [tex]\(a\)[/tex] is [tex]\(a^2\)[/tex].
3. Simplify the variable [tex]\(b\)[/tex]:
The numerator has [tex]\(b^5\)[/tex] and the denominator has [tex]\(b^{-6}\)[/tex].
Again, using the properties of exponents,
[tex]\[
\frac{b^5}{b^{-6}} = b^{5 - (-6)} = b^{5 + 6} = b^{11}
\][/tex]
So, the simplified expression involving [tex]\(b\)[/tex] is [tex]\(b^{11}\)[/tex].
4. Combine all parts:
Combining the simplified coefficient, [tex]\(a\)[/tex] term, and [tex]\(b\)[/tex] term, the simplified expression is:
[tex]\[
\frac{3}{2} \cdot a^2 \cdot b^{11} = \frac{3 a^2 b^{11}}{2}
\][/tex]
Therefore, the equivalent expression 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]
