Let's simplify the given expression step by step:

The expression is:
$\left(a^{-1} \cdot a^{-2}\right)^{-1}$

Step 1: Simplify the base inside the parentheses

Using the property of exponents $a^m \cdot a^n = a^{m+n}$, we combine the exponents of $a^{-1}$ and $a^{-2}$: $a^{-1} \cdot a^{-2} = a^{-1 + (-2)} = a^{-3}$

So, the expression becomes: $\left(a^{-3}\right)^{-1}$

Step 2: Apply the power of a power rule

Using the property $\left(a^m\right)^n = a^{m \cdot n}$, we simplify $\left(a^{-3}\right)^{-1}$: $\left(a^{-3}\right)^{-1} = a^{-3 \cdot (-1)} = a^3$

Final Answer:

$a^3$

Would you like a deeper explanation or have any questions? 😊

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Related Questions:

1. What are the rules of exponents for simplifying products and powers?
2. How do you simplify expressions involving negative exponents?
3. How would the expression change if it were $\left(a^{-1} \cdot a^{-2}\right)^2$?
4. Can you explain the difference between $a^{-1}$ and $1/a$?
5. What happens if the base $a$ is a fraction in this problem?

Tip:

Always simplify the base of an exponentiation first before applying the power rule—it keeps calculations more manageable!