Let's simplify the expression \(\left(\frac{2 a b}{a^{-5} b^2}\right)^{-3}\) step by step.
First, simplify the fraction inside the parentheses:
[tex]\[
\frac{2 a b}{a^{-5} b^2}
\][/tex]
To simplify this expression, recognize that \(a^{-5}\) is the same as \(\frac{1}{a^5}\). Therefore, we can rewrite the fraction as:
[tex]\[
\frac{2 a b}{\frac{1}{a^5} b^2} = 2 a b \cdot \frac{a^5}{b^2}
\][/tex]
Next, combine the terms by using the properties of exponents:
[tex]\[
2 a b \cdot a^5 \cdot \frac{1}{b^2} = 2 a^{1+5} b^{1-2} = 2 a^6 b^{-1}
\][/tex]
So, we have:
[tex]\[
\frac{2 a b}{a^{-5} b^2} = 2 a^6 b^{-1}
\][/tex]
Raise this expression to the power of \(-3\):
[tex]\[
(2 a^6 b^{-1})^{-3}
\][/tex]
Use the properties of exponents to distribute the exponentiation:
[tex]\[
(2^{-3}) (a^6)^{-3} (b^{-1})^{-3} = \frac{1}{2^3} \cdot a^{-18} \cdot b^3 = \frac{b^3}{8 a^{18}}
\][/tex]
Thus, the simplified form of \(\left(\frac{2 a b}{a^{-5} b^2}\right)^{-3}\) is:
[tex]\[
\boxed{\frac{b^3}{8 a^{18}}}
\][/tex]
