To solve the given expression [tex]\(\left(3 m^{-4}\right)^3 \left(3 m^5\right)\)[/tex], we can follow these steps:
1. First, simplify the expression [tex]\(\left(3 m^{-4}\right)^3\)[/tex]:
The expression [tex]\(\left(3 m^{-4}\right)^3\)[/tex] can be broken down using the properties of exponents. Specifically, we apply the power rule [tex]\((ab)^n = a^n b^n\)[/tex]:
[tex]\[
\left(3 m^{-4}\right)^3 = 3^3 \cdot (m^{-4})^3
\][/tex]
2. Calculate [tex]\(3^3\)[/tex]:
[tex]\[
3^3 = 27
\][/tex]
3. Calculate [tex]\((m^{-4})^3\)[/tex]:
Apply the power rule for exponents [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(m^{-4})^3 = m^{-4 \cdot 3} = m^{-12}
\][/tex]
4. Combine the results:
[tex]\[
\left(3 m^{-4}\right)^3 = 27 m^{-12}
\][/tex]
5. Now multiply this result by [tex]\(3 m^5\)[/tex]:
[tex]\[
27 m^{-12} \cdot 3 m^5
\][/tex]
6. Combine the coefficients (number parts):
[tex]\[
27 \cdot 3 = 81
\][/tex]
7. Combine the powers of [tex]\(m\)[/tex]:
When multiplying expressions with the same base, we add the exponents:
[tex]\[
m^{-12} \cdot m^5 = m^{-12 + 5} = m^{-7}
\][/tex]
8. Put it all together:
[tex]\[
81 m^{-7}
\][/tex]
9. Write the expression with positive exponents:
Using the negative exponent rule [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]:
[tex]\[
81 m^{-7} = \frac{81}{m^7}
\][/tex]
10. Match this with the given choices:
A. [tex]\(\frac{81}{m^2}\)[/tex]
B. [tex]\(\frac{27}{m^7}\)[/tex]
C. [tex]\(\frac{27}{m^2}\)[/tex]
D. [tex]\(\frac{81}{m^7}\)[/tex]
The correct answer that matches our result is:
D. [tex]\(\frac{81}{m^7}\)[/tex]
