Which is the largest among the numbers `sqrt(5)`, `3sqrt(7)`, `4sqrt(13)`

Options

A. `sqrt(5)`
B. `3sqrt(7)`
C. `4sqrt(13)`
D. All are equal

Answer

To determine which of the numbers \( \sqrt{5} \), \( 3\sqrt{7} \), and \( 4\sqrt{13} \) is the largest, we can compare their squares to avoid dealing with square roots directly.

### Step-by-Step Solution:

1. **Calculate the square of each number:**
- For \( \sqrt{5} \):
\[ (\sqrt{5})^2 = 5 \]
- For \( 3\sqrt{7} \):
\[ (3\sqrt{7})^2 = 3^2 \cdot (\sqrt{7})^2 = 9 \cdot 7 = 63 \]
- For \( 4\sqrt{13} \):
\[ (4\sqrt{13})^2 = 4^2 \cdot (\sqrt{13})^2 = 16 \cdot 13 = 208 \]

2. **Compare the squared values:**
- We have:
- \( \sqrt{5}^2 = 5 \)
- \( 3\sqrt{7}^2 = 63 \)
- \( 4\sqrt{13}^2 = 208 \)

3. **Identify the largest squared value:**
- Comparing the results:
- \( 5 < 63 < 208 \)

4. **Conclusion:**
- Since \( 208 \) is the largest squared value, it follows that \( 4\sqrt{13} \) is the largest among the three numbers.

### Final Answer:
The largest number among \( \sqrt{5} \), \( 3\sqrt{7} \), and \( 4\sqrt{13} \) is \( 4\sqrt{13} \).