Question

simplify.
\\(\frac{4m^{4}}{3m^{7}}\\)^{-3}
write your answer using only positive exponents.

Explanation:

Step1: Simplify the fraction inside the parentheses

First, use the quotient - rule of exponents $\frac{a^m}{a^n}=a^{m - n}$. For $\frac{4m^{4}}{3m^{7}}$, we have $\frac{4}{3}m^{4-7}=\frac{4}{3}m^{-3}$.
So the original expression becomes $(\frac{4}{3}m^{-3})^{-3}$.

Step2: Apply the power - of - a - product rule $(ab)^n=a^n b^n$

$(\frac{4}{3}m^{-3})^{-3}=(\frac{4}{3})^{-3}(m^{-3})^{-3}$.

Step3: Calculate $(\frac{4}{3})^{-3}$ and $(m^{-3})^{-3}$

For a non - zero number $a$ and integer $n$, $a^{-n}=\frac{1}{a^{n}}$. So $(\frac{4}{3})^{-3}=\frac{3^{3}}{4^{3}}=\frac{27}{64}$.
Using the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, $(m^{-3})^{-3}=m^{(-3)\times(-3)} = m^{9}$.

Step4: Combine the results

$(\frac{4}{3})^{-3}(m^{-3})^{-3}=\frac{27m^{9}}{64}$.

Answer:

$\frac{27m^{9}}{64}$