https://r2.erweima.ai/mathbot/image/1726793896593h793advl.jpg

The expression you've uploaded is:

\[
\left( \frac{m^{-2}}{4m^{-7}} \right)^{-3}
\]

Let's simplify it step by step.

### Step 1: Simplify the inner fraction

First, simplify the fraction inside the parentheses:

\[
\frac{m^{-2}}{4m^{-7}} = \frac{m^{-2}}{4} \cdot \frac{1}{m^{-7}} = \frac{m^{-2}}{4} \cdot m^7
\]

Now simplify the powers of \(m\):

\[
\frac{m^{-2} \cdot m^7}{4} = \frac{m^{5}}{4}
\]

So, the expression inside the parentheses becomes:

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

### Step 2: Apply the exponent \(-3\)

Now apply the exponent \(-3\) to both the numerator and the denominator:

\[
\left( \frac{m^5}{4} \right)^{-3} = \frac{(m^5)^{-3}}{4^{-3}} = \frac{m^{-15}}{4^{-3}} = \frac{m^{-15}}{1/64}
\]

### Step 3: Simplify the expression

Multiply by the reciprocal of \(4^{-3}\):

\[
\frac{m^{-15}}{1/64} = 64m^{-15}
\]

### Final Answer:

\[
64m^{-15}
\]

Would you like further clarification or details on this solution?

Here are 5 related questions to expand your understanding:
1. How do you simplify expressions involving negative exponents?
2. What is the rule for raising a fraction to a negative power?
3. How do you apply the laws of exponents to simplify products?
4. How does multiplying by the reciprocal work in fraction simplification?
5. Can you explain how the final result changes if a variable has a positive exponent instead?

**Tip:** When raising a fraction to a negative exponent, remember to flip the fraction and change the sign of the exponent.

Simplify the expression (m^-2 / 4m^-7)^-3

Learn how to simplify the expression (m^-2 / 4m^-7)^-3 step by step, applying the laws of exponents, fraction simplification, and the reciprocal rule. This solution is suitable for students in Grades 9-12.

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