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Let's go through each part of this problem step-by-step.

### Part (a): Applying the Product of Powers Rule

To evaluate \(\left(3^{\frac{1}{2}}\right)^2\), the problem uses the **Product of Powers Rule**, which states:
\[
(a^m)^n = a^{m \cdot n}
\]
1. We have \(\left(3^{\frac{1}{2}}\right)^2 = 3^{\frac{1}{2} \cdot 2}\).
2. Simplifying \(\frac{1}{2} \cdot 2\) gives \(1\).
3. Therefore, \(\left(3^{\frac{1}{2}}\right)^2 = 3^1 = 3\).

So, the blanks should be filled as follows:
- The first blank should be \( \frac{1}{2} \).
- The second blank should be \( \frac{1}{2} \).
- The third blank should be \(1\).

### Part (b): Defining \(3^{\frac{1}{2}}\)

To ensure consistency with the Product of Powers Rule, \(3^{\frac{1}{2}}\) should be defined as the square root of 3:
\[
3^{\frac{1}{2}} = \sqrt{3}
\]

Therefore, the correct answer in part (b) is:
- \(\sqrt{3}\)

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Would you like a deeper explanation or have any questions?

### Relative Questions
1. What is the Product of Powers Rule in exponents?
2. How can fractional exponents be interpreted in terms of roots?
3. Why does \(3^{\frac{1}{2}} = \sqrt{3}\) make sense mathematically?
4. Can fractional exponents represent cube roots and higher roots?
5. How do we apply the rules of exponents with fractional exponents?

### Tip:
Remember, a fractional exponent like \(\frac{1}{2}\) corresponds to taking the square root, while \(\frac{1}{3}\) corresponds to taking the cube root, and so forth.

Complete the parts below to define 3^(1/2).

Explore how to define 3^(1/2) by applying the Product of Powers Rule. This step-by-step explanation shows how fractional exponents correspond to roots and why 3^(1/2) equals √3, perfect for high school students studying exponents and powers.

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