Let's simplify the expression:
[tex]\[
\frac{a^{-5} b^{-4}}{a^{-13} b^8}
\][/tex]
We start by using the properties of exponents. The properties of exponents tell us that when we divide powers with the same base, we subtract the exponents.
First, simplify the [tex]\(a\)[/tex] terms:
[tex]\[
\frac{a^{-5}}{a^{-13}} = a^{-5 - (-13)} = a^{-5 + 13} = a^8
\][/tex]
Next, simplify the [tex]\(b\)[/tex] terms:
[tex]\[
\frac{b^{-4}}{b^8} = b^{-4 - 8} = b^{-12}
\][/tex]
So, combining these results, we get:
[tex]\[
\frac{a^{-5} b^{-4}}{a^{-13} b^8} = a^8 b^{-12}
\][/tex]
Since [tex]\(b^{-12} = \frac{1}{b^{12}}\)[/tex], we can write the expression as:
[tex]\[
\frac{a^8}{b^{12}}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{a^8}{b^{12}}}
\][/tex]
