https://r2.erweima.ai/mathbot/image/1726981753093izrnwrf2.jpg

Let's go through the problems from the image one by one, simplifying each expression.

---

### **Problem 8 (a)**

\[
\frac{\left( m^3 n \right)^6}{m^{10} n^{-4}}
\]

#### Step-by-step solution:

1. Expand the numerator using exponent rules:
   \[
   \left( m^3 n \right)^6 = m^{3 \cdot 6} n^{1 \cdot 6} = m^{18} n^6
   \]
   So the expression becomes:
   \[
   \frac{m^{18} n^6}{m^{10} n^{-4}}
   \]

2. Simplify the exponents:
   \[
   \frac{m^{18}}{m^{10}} = m^{18 - 10} = m^8
   \]
   \[
   \frac{n^6}{n^{-4}} = n^{6 - (-4)} = n^{6 + 4} = n^{10}
   \]

   Final result:
   \[
   m^8 n^{10}
   \]

---

### **Problem 8 (b)**

\[
\frac{8x^{-4} y}{\left( 2x^2 y^{-1} \right)^2}
\]

#### Step-by-step solution:

1. Simplify the denominator:
   \[
   \left( 2x^2 y^{-1} \right)^2 = 2^2 x^{2 \cdot 2} y^{-1 \cdot 2} = 4x^4 y^{-2}
   \]

2. Now the expression becomes:
   \[
   \frac{8x^{-4} y}{4x^4 y^{-2}} = \frac{8}{4} \cdot \frac{x^{-4}}{x^4} \cdot \frac{y}{y^{-2}}
   \]

3. Simplify each part:
   \[
   \frac{8}{4} = 2
   \]
   \[
   \frac{x^{-4}}{x^4} = x^{-4 - 4} = x^{-8}
   \]
   \[
   \frac{y}{y^{-2}} = y^{1 - (-2)} = y^{1 + 2} = y^3
   \]

   Final result:
   \[
   2x^{-8} y^3
   \]

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Would you like me to continue with the remaining questions? Let me know if you have any questions so far!

### Related Questions:
1. How can I simplify a quotient of powers with negative exponents?
2. What is the rule for raising a product of variables to an exponent?
3. How do we handle exponents when simplifying expressions with multiple variables?
4. Why is the exponent zero rule important in simplifying expressions?
5. How can I simplify a product of terms that involves both positive and negative exponents?

**Tip**: When simplifying expressions, always deal with exponents separately for each base.

Simplify the algebraic expressions (m^3n)^6 / m^10n^-4 and 8x^-4y / (2x^2y^-1)^2.

Learn how to simplify algebraic expressions using exponent rules. This problem covers simplifying (m^3n)^6 / m^10n^-4 and 8x^-4y / (2x^2y^-1)^2 using the laws of exponents, including negative exponents. Suitable for students in Grades 9-11.

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