Which one is greatest out of `root(3)(5), root(4)(3), root(5)(4)` ?

Answer

To determine which of the three numbers \( \sqrt[3]{5} \), \( \sqrt[4]{3} \), and \( \sqrt[5]{4} \) is the greatest, we can compare them by expressing them with a common exponent.

### Step-by-step Solution:

1. **Express Each Root with a Common Exponent**:
To compare the numbers, we can express each root with a common exponent. The least common multiple (LCM) of the denominators (3, 4, and 5) is 60. Therefore, we will express each term as follows:
- \( \sqrt[3]{5} = 5^{1/3} = 5^{20/60} \)
- \( \sqrt[4]{3} = 3^{1/4} = 3^{15/60} \)
- \( \sqrt[5]{4} = 4^{1/5} = 4^{12/60} \)

2. **Rewrite Each Expression**:
Now we rewrite each expression:
- \( \sqrt[3]{5} = 5^{20/60} \)
- \( \sqrt[4]{3} = 3^{15/60} \)
- \( \sqrt[5]{4} = 4^{12/60} \)

3. **Calculate Each Expression**:
To compare, we will calculate the values of \( 5^{20/60} \), \( 3^{15/60} \), and \( 4^{12/60} \):
- \( 5^{20/60} = (5^{1/3})^{20} \)
- \( 3^{15/60} = (3^{1/4})^{15} \)
- \( 4^{12/60} = (4^{1/5})^{12} \)

4. **Estimate the Values**:
We can estimate the values:
- \( 5^{20/60} \approx 2.924 \)
- \( 3^{15/60} \approx 2.279 \)
- \( 4^{12/60} \approx 2.297 \)

5. **Comparison**:
Now we compare the estimated values:
- \( 5^{20/60} \approx 2.924 \) (greatest)
- \( 3^{15/60} \approx 2.279 \)
- \( 4^{12/60} \approx 2.297 \)

6. **Conclusion**:
Therefore, the greatest number among \( \sqrt[3]{5} \), \( \sqrt[4]{3} \), and \( \sqrt[5]{4} \) is \( \sqrt[3]{5} \).

### Final Answer:
The greatest number is \( \sqrt[3]{5} \).