Let's analyze the expressions shown in the image and solve the problem step by step.

The question asks to find all expressions equivalent to $9^4 \cdot 9^{-7}$.

Simplifying $9^4 \cdot 9^{-7}$:

We apply the property of exponents:
$a^m \cdot a^n = a^{m+n}$

Thus:
$9^4 \cdot 9^{-7} = 9^{4 + (-7)} = 9^{-3}$

So, the expression simplifies to $9^{-3}$, or $\frac{1}{9^3}$.

Now, let's compare this result with the provided options:

1. $\frac{9^5}{9^8}$:
   This simplifies using the rule $\frac{a^m}{a^n} = a^{m-n}$:
   $\frac{9^5}{9^8} = 9^{5-8} = 9^{-3}$
   This is equivalent to $9^{-3}$, so this option is correct.

2. $\frac{1}{9^3}$:
   This is exactly $9^{-3}$, so this option is correct.

3. $(9^4)^{-7}$:
   This simplifies as $(a^m)^n = a^{m \cdot n}$:
   $(9^4)^{-7} = 9^{4 \cdot (-7)} = 9^{-28}$
   This is not equivalent to $9^{-3}$, so this option is incorrect.

4. $9^{-28}$:
   This is not equivalent to $9^{-3}$, so this option is incorrect.

Correct answers:

• $\frac{9^5}{9^8}$
• $\frac{1}{9^3}$

Would you like more details or have any questions?

Here are five questions to further explore this concept:

1. What is the general formula for multiplying exponents with the same base?
2. How would you handle multiplying exponents with different bases?
3. What does a negative exponent mean mathematically?
4. Can you express $9^{-3}$ as a fraction with a positive exponent?
5. How does the exponent rule work when dividing two numbers with exponents?

Tip: When dealing with exponents, always try to simplify the expression by combining exponents using basic exponent rules like addition and subtraction.