Question

select the correct answer.
which expression is equivalent to the given expression?
$(3m^{-4})^3(3m^5)$
a. $\frac{81}{m^7}$
b. $\frac{27}{m^7}$
c. $\frac{27}{m^3}$
d. $\frac{81}{m^7}$

Explanation:

Step1: Simplify \((3m^{-4})^{3}\)

Using the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n = a^{mn}\), we have:
\((3m^{-4})^{3}=3^{3}\times(m^{-4})^{3}=27\times m^{-12}=27m^{-12}\)

Step2: Multiply by \(3m^{5}\)

Now, multiply \(27m^{-12}\) by \(3m^{5}\). Using the product rule for exponents \(a^m\times a^n = a^{m + n}\) and the product of coefficients:
\(27m^{-12}\times3m^{5}=(27\times3)\times m^{-12 + 5}=81m^{-7}\)

Step3: Rewrite with positive exponent

Recall that \(a^{-n}=\frac{1}{a^{n}}\), so \(81m^{-7}=\frac{81}{m^{7}}\)

Answer:

D. \(\frac{81}{m^{7}}\)