Let's start simplifying the given expression [tex]\(\left(3 m^{-4}\right)^3\left(3 m^5\right)\)[/tex].
1. Simplify [tex]\(\left(3 m^{-4}\right)^3\)[/tex]:
- First, we raise both the coefficient and the variable with its exponent to the power of 3.
- For the coefficient: [tex]\(3^3 = 27\)[/tex]
- For the variable: [tex]\((m^{-4})^3\)[/tex]
[tex]\[
(m^{-4})^3 = m^{-4 \cdot 3} = m^{-12}
\][/tex]
- Combining these, we get:
[tex]\[
\left(3 m^{-4}\right)^3 = 27 m^{-12}
\][/tex]
2. Multiplying [tex]\(27 m^{-12}\)[/tex] by [tex]\(3 m^5\)[/tex]:
- Multiply the coefficients: [tex]\(27 \cdot 3 = 81\)[/tex]
- For the exponents of [tex]\(m\)[/tex], we use the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
m^{-12} \cdot m^5 = m^{-12 + 5} = m^{-7}
\][/tex]
3. Combine the results:
- Thus, the product is:
[tex]\[
81 m^{-7}
\][/tex]
- Rewriting [tex]\(m^{-7}\)[/tex] as a fraction, we get:
[tex]\[
81 m^{-7} = \frac{81}{m^7}
\][/tex]
Therefore, the simplified expression is [tex]\(\frac{81}{m^7}\)[/tex].
The correct answer is:
[tex]\[
\boxed{D}
\][/tex]
