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.