https://file.aiquickdraw.com/mathbot/file/17265648189176q885t99.jpg

The expression provided in the image is:

\[
(2a^{-2}b^0)^3
\]

Let's simplify this step by step:

1. **Simplify powers within the parentheses**:
    - \( b^0 = 1 \) (since any number raised to the power of 0 is 1).
    - So the expression inside the parentheses simplifies to:
      \[
      (2a^{-2} \cdot 1) = 2a^{-2}
      \]

2. **Raise everything inside the parentheses to the power of 3**:
    - Apply the exponent to each term inside the parentheses:
      \[
      (2a^{-2})^3 = 2^3 \cdot (a^{-2})^3
      \]
    - This gives:
      \[
      2^3 \cdot a^{-6} = 8a^{-6}
      \]

So, the simplified expression is:
\[
8a^{-6}
\]

Let me know if you'd like more details or have any questions!

Here are five related questions to consider:
1. What is the general rule for exponents when raising a power to a power?
2. How do you simplify expressions with negative exponents?
3. Why does \( b^0 = 1 \) for any non-zero \( b \)?
4. How would this expression change if the exponent outside the parentheses was different, say 2 instead of 3?
5. What is the significance of \( a^{-6} \) and how would you interpret it in terms of division?

**Tip**: Negative exponents indicate reciprocals. For example, \( a^{-6} \) can be rewritten as \( \frac{1}{a^6} \).

This problem involves simplifying the expression (2a^{-2}b^0)^3 using exponent rules. The solution applies the zero exponent theorem, exponent multiplication rule, and the negative exponent concept. The problem is suitable for students in Grades 8-10, helping them understand how to manage powers and negative exponents.

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