https://file.aiquickdraw.com/mathbot/file/17277195332790n1wgbw4.jpg

The expression shown in the image is:

\[
\frac{9f^{-9}g^0}{9f^0g^3}
\]

Let's simplify this step by step:

1. **Simplify the constants**: 
   - The "9" in the numerator and the denominator can cancel out, so we're left with:
   \[
   \frac{f^{-9}g^0}{f^0g^3}
   \]

2. **Simplify powers of \(g\)**:
   - \(g^0 = 1\), because anything raised to the power of 0 is 1. So the expression simplifies to:
   \[
   \frac{f^{-9}}{f^0g^3}
   \]

3. **Simplify powers of \(f\)**:
   - \(f^0 = 1\), so the expression becomes:
   \[
   \frac{f^{-9}}{g^3}
   \]

4. **Final expression**: 
   - The expression is now:
   \[
   \frac{f^{-9}}{g^3} = \frac{1}{f^9g^3}
   \]

Thus, the simplified form of the expression is:

\[
\frac{1}{f^9g^3}
\]

Would you like any further details or explanations? 

Here are five related questions to explore:
1. What are the general rules for simplifying powers with the same base?
2. How does negative exponentiation work, and how can it be interpreted in terms of fractions?
3. How do you simplify expressions with exponents of 0?
4. What happens when you have multiple variables in a fraction, and how do you simplify each separately?
5. Can you multiply expressions with different bases and still simplify the exponents?

**Tip**: Remember that a negative exponent represents the reciprocal of the base raised to the positive exponent.

Simplify the expression 9f^(-9)g^0 / 9f^0g^3

This solution simplifies the algebraic expression 9f^(-9)g^0 / 9f^0g^3 using rules of exponents, including zero exponent, negative exponents, and simplification. Suitable for students in Grades 8-10.

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