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The question asks: "Which of the following is the largest number?"

The given options are:

- A: \( 1^{-\frac{1}{2}} \)
- B: \( 2^{\frac{1}{2}} \)
- C: \( \sqrt{3} \)
- D: \( 3 \sqrt{5} \)
- E: \( \frac{1}{2} \sqrt{5} \)

Let's evaluate each expression:

1. **A: \( 1^{-\frac{1}{2}} \)**
   \[
   1^{-\frac{1}{2}} = 1
   \]
   Since any power of 1 is still 1, this evaluates to 1.

2. **B: \( 2^{\frac{1}{2}} \)**
   \[
   2^{\frac{1}{2}} = \sqrt{2} \approx 1.414
   \]

3. **C: \( \sqrt{3} \)**
   \[
   \sqrt{3} \approx 1.732
   \]

4. **D: \( 3 \sqrt{5} \)**
   \[
   3 \sqrt{5} = 3 \times \sqrt{5} \approx 3 \times 2.236 = 6.708
   \]

5. **E: \( \frac{1}{2} \sqrt{5} \)**
   \[
   \frac{1}{2} \sqrt{5} = \frac{1}{2} \times 2.236 \approx 1.118
   \]

Now, comparing the values:

- A: 1
- B: 1.414
- C: 1.732
- D: 6.708
- E: 1.118

The largest number is **D: \( 3 \sqrt{5} \)**, approximately 6.708.

Would you like further explanation or have any questions?

Here are 5 related questions to consider:
1. How can we simplify powers of numbers like \( 2^{\frac{1}{2}} \)?
2. What is the importance of comparing square roots in evaluating numbers?
3. How do fractional exponents relate to roots, like in \( 1^{-\frac{1}{2}} \)?
4. Why is \( 3 \sqrt{5} \) larger than other values involving roots?
5. Can you think of other similar problems involving roots and exponents?

**Tip**: When comparing expressions with square roots, it’s helpful to approximate their decimal values to easily identify the largest or smallest.

Which of the following is the largest number?

This problem asks students to compare numbers involving square roots and exponents to determine the largest value. Learn the steps to evaluate expressions like 2^(1/2), √3, and 3√5, and compare their approximate decimal values.

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