https://file.aiquickdraw.com/mathbot/file/1726391706348sgvhxs6m.jpg

Let's work through the expression from your image: 

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

### Step-by-Step Breakdown:
1. **Simplifying \(\left( \sqrt{2} \right)^3\)**:
   - Recall that \(\sqrt{2}\) is the same as \(2^{\frac{1}{2}}\).
   - So, \(\left( \sqrt{2} \right)^3 = \left( 2^{\frac{1}{2}} \right)^3\).
   - Apply the exponent rule \(\left( a^m \right)^n = a^{m \cdot n}\), which gives us \(2^{\frac{1}{2} \cdot 3} = 2^{\frac{3}{2}}\).

2. **Combining with \(2^{-\frac{1}{2}}\)**:
   - Now you have \(2^{\frac{3}{2}} \cdot 2^{-\frac{1}{2}}\).
   - Use the property of exponents \(a^m \cdot a^n = a^{m+n}\).
   - This gives us: 
     \[
     2^{\frac{3}{2} + \left(-\frac{1}{2}\right)} = 2^{\frac{2}{2}} = 2^1.
     \]

### Final Answer:
The simplified expression is \(2\).

So the correct option is the last one: \( \boxed{2} \).

Let me know if you need further explanation or if you have any questions!

Here are 5 related questions you could explore:
1. What is the general rule for multiplying powers with the same base?
2. How would you simplify an expression involving square roots and powers, like \(\left( \sqrt{3} \right)^5\)?
3. Can you apply the same exponent rules for negative bases?
4. How do fractional exponents relate to radicals (e.g., \(\sqrt[3]{8}\))?
5. How do you simplify expressions with exponents that involve division?

**Tip**: Always simplify expressions with the same base by adding or subtracting their exponents according to the operation (multiplication or division).

Using the exponent properties, which of the following expressions is equivalent to (√2)^3 * 2^(-1/2)?

Learn how to simplify the expression (√2)^3 * 2^(-1/2) using exponent properties. This problem covers exponent rules and radicals, suitable for students in Grades 8-10.

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