Which of the following surds is greatest?
`root(3)(2),sqrt3,root(4)(5),root(6)(7)`

Answer

To determine which of the following surds is greatest: \( \sqrt[3]{2}, \sqrt{3}, \sqrt[4]{5}, \sqrt[6]{7} \), we will convert each surd into an exponential form and then compare their values.

### Step-by-Step Solution:

1. **Convert Each Surd to Exponential Form:**
- \( \sqrt[3]{2} = 2^{\frac{1}{3}} \)
- \( \sqrt{3} = 3^{\frac{1}{2}} \)
- \( \sqrt[4]{5} = 5^{\frac{1}{4}} \)
- \( \sqrt[6]{7} = 7^{\frac{1}{6}} \)

2. **Identify a Common Denominator for the Exponents:**
- The denominators of the exponents are 3, 2, 4, and 6. The least common multiple (LCM) of these numbers is 12.

3. **Rewrite Each Expression with the Common Denominator:**
- Convert each exponent to have a denominator of 12:
- \( 2^{\frac{1}{3}} = 2^{\frac{4}{12}} \)
- \( 3^{\frac{1}{2}} = 3^{\frac{6}{12}} \)
- \( 5^{\frac{1}{4}} = 5^{\frac{3}{12}} \)
- \( 7^{\frac{1}{6}} = 7^{\frac{2}{12}} \)

4. **Rewrite Each Surd:**
- Now we can express each surd as:
- \( 2^{\frac{4}{12}} \)
- \( 3^{\frac{6}{12}} \)
- \( 5^{\frac{3}{12}} \)
- \( 7^{\frac{2}{12}} \)

5. **Calculate the Base Values:**
- Now we need to calculate the base values:
- \( 2^4 = 16 \)
- \( 3^6 = 729 \)
- \( 5^3 = 125 \)
- \( 7^2 = 49 \)

6. **Compare the Values:**
- Now we compare the values:
- \( 16^{\frac{1}{12}} \)
- \( 729^{\frac{1}{12}} \)
- \( 125^{\frac{1}{12}} \)
- \( 49^{\frac{1}{12}} \)

7. **Determine the Greatest Value:**
- The greatest base value is \( 729 \) (which corresponds to \( \sqrt{3} \)).
- Therefore, \( \sqrt{3} \) is the greatest surd among the given options.

### Final Answer:
The greatest surd is \( \sqrt{3} \).