To simplify the given expression [tex]\(\left(3 m^{-4}\right)^3 \left(3 m^5\right)\)[/tex], we need to perform the following steps.
1. First, simplify [tex]\(\left(3 m^{-4}\right)^3\)[/tex]:
- By raising both the coefficient and the exponent to the power of 3, we get:
[tex]\[
(3 m^{-4})^3 = 3^3 \cdot (m^{-4})^3 = 27 \cdot m^{-12}
\][/tex]
2. Next, multiply the simplified result by [tex]\(3 m^5\)[/tex]:
- When multiplying coefficients, multiply the numbers:
[tex]\[
27 \cdot 3 = 81
\][/tex]
- And for the exponents of [tex]\(m\)[/tex], add them:
[tex]\[
m^{-12} \cdot m^5 = m^{-12 + 5} = m^{-7}
\][/tex]
3. Combine the results:
[tex]\[
81 \cdot m^{-7}
\][/tex]
4. Express the negative exponent as a fraction:
[tex]\[
81 \cdot m^{-7} = \frac{81}{m^7}
\][/tex]
Therefore, the correct expression is equivalent to [tex]\(\frac{81}{m^7}\)[/tex]. However, none of the provided options matches this simplified result exactly.
Given the choices:
A. [tex]\(\frac{81}{m^2}\)[/tex]
B. [tex]\(\frac{27}{m^7}\)[/tex]
C. [tex]\(\frac{27}{m^2}\)[/tex]
D. [tex]\(\frac{1}{m^7}\)[/tex]
None of these options are correct based on our detailed step-by-step simplification. Nonetheless, our correct simplified expression [tex]\(\frac{81}{m^7}\)[/tex] is the answer derived.
