With out actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion :
(i) `(17)/(8)` (ii) `(64)/(455)` (iii) `(29)/(343)` (iv) `(129)/(2^(2)5^(7)7^(2))` (v) `(6)/(15)` (vi) `(27)/(210)`

Answer

To determine whether the given rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- A rational number \( \frac{p}{q} \) has a terminating decimal expansion if:
- \( p \) and \( q \) are coprime (i.e., their greatest common divisor is 1).
- The denominator \( q \) can be expressed in the form \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.

2. **Analyzing Each Rational Number**:

**(i) \( \frac{17}{8} \)**:
- \( 17 \) and \( 8 \) are coprime (GCD is 1).
- \( 8 = 2^3 \) (only powers of 2).
- **Conclusion**: Terminating decimal expansion.

**(ii) \( \frac{64}{455} \)**:
- \( 64 \) and \( 455 \) are coprime (GCD is 1).
- \( 455 = 5 \times 91 = 5 \times 7 \times 13 \) (contains primes other than 2 and 5).
- **Conclusion**: Non-terminating repeating decimal expansion.

**(iii) \( \frac{29}{343} \)**:
- \( 29 \) and \( 343 \) are coprime (GCD is 1).
- \( 343 = 7^3 \) (only powers of 7).
- **Conclusion**: Non-terminating repeating decimal expansion.

**(iv) \( \frac{129}{2^2 \cdot 5^7 \cdot 7^2} \)**:
- \( 129 \) and \( 2^2 \cdot 5^7 \cdot 7^2 \) are coprime (GCD is 1).
- The denominator has a factor of \( 7 \) (not in the form of \( 2^m \times 5^n \)).
- **Conclusion**: Non-terminating repeating decimal expansion.

**(v) \( \frac{6}{15} \)**:
- Simplifying gives \( \frac{2}{5} \).
- \( 2 \) and \( 5 \) are coprime (GCD is 1).
- \( 5 \) is in the form of \( 5^1 \) (only powers of 5).
- **Conclusion**: Terminating decimal expansion.

**(vi) \( \frac{27}{210} \)**:
- Simplifying gives \( \frac{9}{70} \).
- \( 9 \) and \( 70 \) are coprime (GCD is 1).
- \( 70 = 2 \times 5 \times 7 \) (contains a factor of 7).
- **Conclusion**: Non-terminating repeating decimal expansion.

### Final Answers:
1. \( \frac{17}{8} \) - Terminating
2. \( \frac{64}{455} \) - Non-terminating repeating
3. \( \frac{29}{343} \) - Non-terminating repeating
4. \( \frac{129}{2^2 \cdot 5^7 \cdot 7^2} \) - Non-terminating repeating
5. \( \frac{6}{15} \) - Terminating
6. \( \frac{27}{210} \) - Non-terminating repeating