To solve this problem, we start by simplifying the expression $a^{-9} \cdot a^8$.

Step 1: Simplify the expression using the law of exponents.

The rule for multiplying exponents with the same base is:

$a^m \cdot a^n = a^{m+n}$

Using this, we have:

$a^{-9} \cdot a^8 = a^{-9 + 8} = a^{-1}$

Step 2: Analyze the answer choices.

We are looking for the expression that is not equivalent to $a^{-1}$.

• Option A: $\frac{a^8}{a^9} = a^{8-9} = a^{-1}$, so this is equivalent to $a^{-1}$.
• Option B: $\frac{a^9}{a^8} = a^{9-8} = a^1$, which is not equivalent to $a^{-1}$.
• Option C: $a^{-1}$ is exactly equivalent to $a^{-1}$.
• Option D: $\frac{1}{a}$ is also equivalent to $a^{-1}$, since $\frac{1}{a} = a^{-1}$.

Conclusion:

The expression that is not equivalent to $a^{-9} \cdot a^8$ is Option B: $\frac{a^9}{a^8}$.

Would you like further details on exponent laws, or have any questions about this problem?

Here are 5 related questions for further practice:

1. What is the result of $a^3 \cdot a^5$?
2. How do you simplify $\frac{a^7}{a^4}$?
3. What is $a^0$?
4. Simplify $a^2 \cdot a^{-3}$.
5. How do negative exponents affect the base?

Tip: When multiplying or dividing terms with the same base, always add or subtract the exponents according to the laws of exponents.