To simplify the expression [tex]\(\left(\frac{a^{-3}}{2 b^4}\right)^3\)[/tex], let's break it down step by step.
1. Start with the given expression:
[tex]\[
\left(\frac{a^{-3}}{2 b^4}\right)^3
\][/tex]
2. Apply the power to both the numerator and the denominator separately:
[tex]\[
\left(\frac{a^{-3}}{2 b^4}\right)^3 = \frac{(a^{-3})^3}{(2 b^4)^3}
\][/tex]
3. Raise the numerator [tex]\(a^{-3}\)[/tex] to the power of 3:
[tex]\[
(a^{-3})^3 = a^{-3 \cdot 3} = a^{-9}
\][/tex]
4. Raise the denominator [tex]\(2 b^4\)[/tex] to the power of 3:
[tex]\[
(2 b^4)^3 = 2^3 \cdot (b^4)^3 = 8 \cdot b^{4 \cdot 3} = 8b^{12}
\][/tex]
5. Now combine the results from the numerator and the denominator:
[tex]\[
\frac{a^{-9}}{8 b^{12}}
\][/tex]
6. Rewrite [tex]\(a^{-9}\)[/tex] using the property of negative exponents ([tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]):
[tex]\[
a^{-9} = \frac{1}{a^9}
\][/tex]
7. Substitute this back into the fraction:
[tex]\[
\frac{\frac{1}{a^9}}{8 b^{12}} = \frac{1}{a^9 \cdot 8 b^{12}} = \frac{1}{8 a^9 b^{12}}
\][/tex]
Finally, the simplified expression is:
[tex]\[
\frac{1}{8 a^9 b^{12}}
\][/tex]
So, the correct choice is:
[tex]\(\boxed{\frac{1}{8 a^9 b^{12}}}\)[/tex]
