To simplify the given expression [tex]\(\left(\frac{a^{-8} b}{a^{-5} b^3}\right)^{-3}\)[/tex], we'll go through the steps methodically:
1. Simplify the inner fraction:
[tex]\[
\frac{a^{-8} b}{a^{-5} b^3}
\][/tex]
We can separate this into two fractions:
[tex]\[
\frac{a^{-8}}{a^{-5}} \times \frac{b}{b^3}
\][/tex]
2. Simplify each part using exponent rules:
For the [tex]\(a\)[/tex] terms, use the rule [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]:
[tex]\[
\frac{a^{-8}}{a^{-5}} = a^{-8-(-5)} = a^{-8+5} = a^{-3}
\][/tex]
For the [tex]\(b\)[/tex] terms, using the same rule:
[tex]\[
\frac{b}{b^3} = b^{1-3} = b^{-2}
\][/tex]
So, the expression simplifies to:
[tex]\[
a^{-3} b^{-2}
\][/tex]
3. Apply the outer exponent [tex]\((-3)\)[/tex]:
[tex]\[
\left(a^{-3} b^{-2}\right)^{-3}
\][/tex]
Use the power of a power rule [tex]\((x^m)^n = x^{mn}\)[/tex]:
[tex]\[
(a^{-3})^{-3} = a^{-3 \times -3} = a^{9}
\][/tex]
[tex]\[
(b^{-2})^{-3} = b^{-2 \times -3} = b^{6}
\][/tex]
4. Combine the simplified parts:
[tex]\[
a^9 b^6
\][/tex]
Therefore, the expression [tex]\(\left(\frac{a^{-8} b}{a^{-5} b^3}\right)^{-3}\)[/tex] is equivalent to [tex]\(a^9 b^6\)[/tex].
So the correct answer is:
[tex]\[ \boxed{a^9 b^6} \][/tex]
