To determine which expression is equivalent to the given expression:
[tex]\[
\left(3 m^{-4}\right)^3 \left(3 m^5\right)
\][/tex]
we will perform a step-by-step simplification:
1. Expand the first term [tex]\(\left(3 m^{-4}\right)^3\)[/tex]:
- When an exponent is applied to a product, we apply the exponent to each factor separately:
[tex]\[
\left(3 m^{-4}\right)^3 = 3^3 \left(m^{-4}\right)^3
\][/tex]
- Simplify each part:
[tex]\[
3^3 = 27
\][/tex]
[tex]\[
\left(m^{-4}\right)^3 = m^{-4 \times 3} = m^{-12}
\][/tex]
- Therefore:
[tex]\[
\left(3 m^{-4}\right)^3 = 27 m^{-12}
\][/tex]
2. Combine the result with the second term [tex]\(3 m^5\)[/tex]:
[tex]\[
\left(27 m^{-12}\right) \left(3 m^5\right)
\][/tex]
3. Simplify the expression by combining the coefficients and the exponents of [tex]\(m\)[/tex]:
- Multiply the coefficients:
[tex]\[
27 \times 3 = 81
\][/tex]
- Add the exponents of [tex]\(m\)[/tex]:
[tex]\[
m^{-12 + 5} = m^{-7}
\][/tex]
- Therefore:
[tex]\[
27 m^{-12} \cdot 3 m^5 = 81 m^{-7}
\][/tex]
4. Rewrite [tex]\(81 m^{-7}\)[/tex] in a different form (with a positive exponent):
[tex]\[
81 m^{-7} = \frac{81}{m^7}
\][/tex]
Thus, the expression [tex]\( \frac{81}{m^7} \)[/tex] is equivalent to the given expression [tex]\(\left(3 m^{-4}\right)^3 \left(3 m^5\right)\)[/tex].
Therefore, the correct choice is:
[tex]\[
\boxed{D. \frac{81}{m^7}}
\][/tex]
