Among `sqrt(2)`, `root3(3)`, `root4(5)`, `root3(2)` which one is the greatest ?

Options

A. `root4(5)`
B. `sqrt(2)`
C. `root3(3)`
D. `root3(2)`

Answer

To determine which among \( \sqrt{2} \), \( \sqrt[3]{3} \), \( \sqrt[4]{5} \), and \( \sqrt[3]{2} \) is the greatest, we can express each of these roots in terms of powers of their bases.

### Step-by-Step Solution:

1. **Express each root in exponent form**:
- \( \sqrt{2} = 2^{1/2} \)
- \( \sqrt[3]{3} = 3^{1/3} \)
- \( \sqrt[4]{5} = 5^{1/4} \)
- \( \sqrt[3]{2} = 2^{1/3} \)

2. **Identify the denominators of the exponents**:
- The denominators are 2, 3, 4, and 3. To compare these numbers, we need a common denominator. The least common multiple (LCM) of 2, 3, and 4 is 12.

3. **Convert each exponent to have the common denominator**:
- Multiply the exponent of each expression by the LCM (12):
- \( \sqrt{2} = 2^{1/2} = 2^{6/12} \)
- \( \sqrt[3]{3} = 3^{1/3} = 3^{4/12} \)
- \( \sqrt[4]{5} = 5^{1/4} = 5^{3/12} \)
- \( \sqrt[3]{2} = 2^{1/3} = 2^{4/12} \)

4. **Now we can compare the values**:
- We have:
- \( 2^{6/12} = 2^{0.5} \)
- \( 3^{4/12} = 3^{0.3333} \)
- \( 5^{3/12} = 5^{0.25} \)
- \( 2^{4/12} = 2^{0.3333} \)

5. **Calculate the values**:
- \( 2^{6/12} = 2^{0.5} = \sqrt{2} \approx 1.414 \)
- \( 3^{4/12} = 3^{0.3333} \approx 1.442 \)
- \( 5^{3/12} = 5^{0.25} \approx 1.495 \)
- \( 2^{4/12} = 2^{0.3333} \approx 1.260 \)

6. **Identify the greatest value**:
- Comparing the approximate values:
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt[3]{3} \approx 1.442 \)
- \( \sqrt[4]{5} \approx 1.495 \)
- \( \sqrt[3]{2} \approx 1.260 \)
- The greatest value is \( \sqrt[4]{5} \).

### Conclusion:
The greatest among \( \sqrt{2} \), \( \sqrt[3]{3} \), \( \sqrt[4]{5} \), and \( \sqrt[3]{2} \) is \( \sqrt[4]{5} \).