https://file.aiquickdraw.com/mathbot/file/1728985593529e5y3ah9l.jpg

The problem in the image asks for the simplification of the expression:

\[
\left( \frac{a^{-3}}{a^{-2}} \right)^{-\frac{1}{2}}
\]

### Step-by-step solution:

1. **Simplify the fraction** inside the parentheses:
   \[
   \frac{a^{-3}}{a^{-2}} = a^{-3 - (-2)} = a^{-3 + 2} = a^{-1}
   \]

2. Now the expression is:
   \[
   (a^{-1})^{-\frac{1}{2}}
   \]

3. Apply the power rule \((x^m)^n = x^{m \cdot n}\):
   \[
   a^{-1 \cdot -\frac{1}{2}} = a^{\frac{1}{2}}
   \]

Thus, the simplified expression is \( a^{\frac{1}{2}} \).

### The correct answer is: **B. \( a^{1/2} \)**

Would you like more details on any of these steps or have any other questions?

Here are 5 related questions to expand on this topic:
1. What is the general rule for simplifying expressions with negative exponents?
2. How would the simplification change if the exponents were positive initially?
3. What happens when an expression with fractional exponents is raised to a negative power?
4. How do you simplify a more complex expression like \(\left(\frac{a^m}{a^n}\right)^p\)?
5. How can we interpret fractional exponents geometrically?

**Tip:** When dealing with exponents, remember that dividing powers of the same base is equivalent to subtracting the exponents.

Let a be a positive real number. Simplify the expression: (a^(-3) / a^(-2))^(-1/2).

This problem demonstrates how to simplify an expression with negative exponents: (a^(-3) / a^(-2))^(-1/2). Learn the steps for applying exponent rules to simplify the expression to its final form.

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