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Select the correct answer.

Which expression is equivalent to the given expression?

[tex]\left(3 m^{-4}\right)^3\left(3 m^5\right)[/tex]

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]


Sagot :

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]