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This is a math worksheet where the task is to simplify expressions and match them with the correct answers. Here's the breakdown of the given expressions and their corresponding simplified forms:

1. \( (x^3)^2 = x^{3 \times 2} = x^6 \) → Corresponds to "H".
2. \( (x^4)^3 = x^{4 \times 3} = x^{12} \) → Corresponds to "G".
3. \( (2x)^3 = 2^3 \cdot x^3 = 8x^3 \) → Corresponds to "S".
4. \( (-4x)^2 = (-4)^2 \cdot x^2 = 16x^2 \) → Corresponds to "P".
5. \( (-3x)^4 = (-3)^4 \cdot x^4 = 81x^4 \) → Corresponds to "L".
6. \( (8x^5)^2 = 8^2 \cdot x^{5 \times 2} = 64x^{10} \) → Corresponds to "E".
7. \( (-2x^3)^5 = (-2)^5 \cdot x^{3 \times 5} = -32x^{15} \) → Corresponds to "T".
8. \( (4x)^3 = 4^3 \cdot x^3 = 64x^3 \) → Corresponds to "I".
9. \( (-9x)^2 = (-9)^2 \cdot x^2 = 81x^2 \) → Corresponds to "L".
10. \( x(2x^2)^3 = x \cdot 2^3 \cdot x^{2 \times 3} = 8x^7 \) → Corresponds to "G".
11. \( -3x(2x)^2 = -3x \cdot 2^2 \cdot x^2 = -12x^3 \) → Corresponds to "N".
12. \( x^2(5x^3)^3 = x^2 \cdot 5^3 \cdot x^{3 \times 3} = 125x^{11} \) → Corresponds to "T".
13. \( -4x^2(-4x)^2 = -4x^2 \cdot 16x^2 = -64x^4 \) → Corresponds to "E".

If you need any clarification or more details, feel free to ask!

### Follow-up Questions:
1. Can you explain how the laws of exponents work in simplifying expressions?
2. What happens if we have negative exponents in these expressions?
3. How does distributing powers over products affect terms like \( (2x)^3 \)?
4. Could we get different results if we change the order of operations in the exponents?
5. Why is the sign important when raising negative numbers to even or odd powers?

### Tip:
Always remember that raising a negative number to an even power makes it positive, but raising it to an odd power keeps it negative.

What Happens to a Dog Who Solves These Math Expressions?

This worksheet involves simplifying algebraic expressions with exponents using the laws of exponents. It covers key concepts such as power of a power and power of a product rules. Suitable for students in Grades 8-10.

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