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### 2. \( b^4 \cdot b^3 \cdot b^2 \)
Using the product of powers rule: \( a^m \cdot a^n = a^{m+n} \), we can add the exponents.

\[
b^4 \cdot b^3 \cdot b^2 = b^{4+3+2} = b^9
\]

### 5. \( (3a)^4 \)
We apply the power to both the constant and the variable. The general rule is \( (ab)^n = a^n b^n \).

\[
(3a)^4 = 3^4 \cdot a^4 = 81a^4
\]

### 8. \( (x^3 y^{-3})^3 \)
Use the power of a product rule: \( (ab)^n = a^n \cdot b^n \), and the power of a power rule: \( (a^m)^n = a^{m \cdot n} \).

\[
(x^3 y^{-3})^3 = x^{3 \cdot 3} \cdot y^{-3 \cdot 3} = x^9 \cdot y^{-9} = \frac{x^9}{y^9}
\]

### 11. \( (4e^{-4})(ac^3)(3a^3 e) \)
We distribute and multiply the constants and variables separately. Use the product of powers rule when multiplying like terms.

\[
(4e^{-4})(ac^3)(3a^3 e) = 4 \cdot 3 \cdot a^1 \cdot a^3 \cdot c^3 \cdot e^{-4} \cdot e^1 = 12 \cdot a^{1+3} \cdot c^3 \cdot e^{-4+1}
\]
\[
= 12a^4c^3e^{-3} = \frac{12a^4c^3}{e^3}
\]

These are the solutions. Do you have any questions or would you like more details on any step?

### Expansion questions:
1. What is the general rule for multiplying exponents with the same base?
2. How do you apply the power rule to a product like \( (xy)^n \)?
3. How do you handle negative exponents in expressions like \( y^{-3} \)?
4. What is the difference between product of powers and power of a power rules?
5. Can you simplify an expression like \( (2x^2 y^{-1})^4 \)?

### Tip:
When multiplying terms with exponents, always check if they have the same base to apply the product of powers rule.

Solve the following algebraic expressions: b^4 * b^3 * b^2, (3a)^4, (x^3 y^-3)^3, (4e^-4)(ac^3)(3a^3 e)

Learn how to simplify expressions like b^4 * b^3 * b^2, (3a)^4, (x^3 y^-3)^3, and (4e^-4)(ac^3)(3a^3 e) using exponentiation rules. This step-by-step solution covers key concepts such as the product of powers, power of a product, and handling negative exponents. Suitable for Grades 9-10.

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