State, with reason, which of the following are surds and which are not :
(i) `sqrt(180) " (ii) "root(3)(64)" (iii) "root(3)(25) times root(3)(40)`

Answer

To determine which of the given expressions are surds and which are not, we need to apply the definition of a surd. A surd is an expression that cannot be simplified to remove the root. Specifically, a number is a surd if it is in the form \( \sqrt[n]{x} \) where \( x \) is not a perfect nth power.

Let's analyze each part step-by-step:

### Step 1: Analyze \( \sqrt{180} \)

1. **Find the prime factorization of 180**:
- \( 180 = 2^2 \times 3^2 \times 5 \)

2. **Check for perfect squares**:
- The factors \( 2^2 \) and \( 3^2 \) are perfect squares, but \( 5 \) is not.

3. **Conclusion**:
- Since \( \sqrt{180} \) cannot be simplified to a whole number (because of the factor \( 5 \)), it is a surd.

### Step 2: Analyze \( \sqrt[3]{64} \)

1. **Find the prime factorization of 64**:
- \( 64 = 4^3 = 2^6 \)

2. **Check for perfect cubes**:
- \( 64 \) is a perfect cube since \( 64 = 4^3 \).

3. **Conclusion**:
- Since \( \sqrt[3]{64} = 4 \) is a whole number, it is not a surd.

### Step 3: Analyze \( \sqrt[3]{25} \times \sqrt[3]{40} \)

1. **Combine the cube roots**:
- \( \sqrt[3]{25} \times \sqrt[3]{40} = \sqrt[3]{25 \times 40} = \sqrt[3]{1000} \)

2. **Find the prime factorization of 1000**:
- \( 1000 = 10^3 \)

3. **Check for perfect cubes**:
- Since \( 1000 \) is a perfect cube, \( \sqrt[3]{1000} = 10 \).

4. **Conclusion**:
- Since \( \sqrt[3]{1000} = 10 \) is a whole number, it is not a surd.

### Final Summary

- \( \sqrt{180} \) is a surd.
- \( \sqrt[3]{64} \) is not a surd.
- \( \sqrt[3]{25} \times \sqrt[3]{40} \) is not a surd.