To simplify the expression $\left( \frac{4m^3n^8}{2m^5n^6} \right)^{-3}$, let's break it down step by step.

Step 1: Simplify the expression inside the parentheses

$\frac{4m^3n^8}{2m^5n^6}$

• For the constants: $\frac{4}{2} = 2$
• For the powers of $m$: $\frac{m^3}{m^5} = m^{3-5} = m^{-2}$
• For the powers of $n$: $\frac{n^8}{n^6} = n^{8-6} = n^2$

So, the simplified expression inside the parentheses is: $2m^{-2}n^2$

Step 2: Apply the negative exponent

Now, raise the entire expression $2m^{-2}n^2$ to the power of $-3$: $\left(2m^{-2}n^2\right)^{-3}$

• For the constant: $2^{-3} = \frac{1}{8}$
• For $m^{-2}$: $\left(m^{-2}\right)^{-3} = m^{6}$ (Remember: $(a^m)^n = a^{m \cdot n}$)
• For $n^2$: $\left(n^2\right)^{-3} = n^{-6}$

Thus, the expression becomes: $\frac{1}{8} m^6 n^{-6}$

Step 3: Final simplified expression

The final expression is: $\frac{m^6}{8n^6}$

This is the simplified form of the original expression.

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Would you like more details or have any questions? Here are some related questions you might find useful:

1. How do you simplify expressions with negative exponents?
2. What happens when you raise a fraction to a negative exponent?
3. How do you handle terms with the same base and different exponents?
4. How can you simplify powers of constants in algebraic expressions?
5. What is the general rule for applying negative exponents to variables?

Tip: When working with exponents, remember that a negative exponent means "take the reciprocal" of the base.