Which of the following rational numbers have terminating decimal expansion ?
(i) `(17)/(8)`
(ii) `(64)/(455)`
(iii) `(15)/(1600)`
(iv) `(13)/(3125)`
(v) `(129)/(2^(2)xx5^(7)xx7^(17))`
(vi) `(987)/(10500)`

Options

A. Only (iv) and (vi)
B. Only (ii) and (v)
C. Only (i),(iii),(iv) and (vi)
D. Only (i), (iii) and (iv)

Answer

To determine which of the given rational numbers have terminating decimal expansions, we need to check the denominators of each fraction. A rational number has a terminating decimal expansion if its denominator (in simplest form) can be expressed as \(2^n \times 5^m\), where \(n\) and \(m\) are non-negative integers.

Let's analyze each option step by step:

### Step 1: Analyze \(\frac{17}{8}\)
- **Denominator**: \(8\)
- **Factorization**: \(8 = 2^3\)
- **Form**: This is in the form \(2^n\) (where \(n=3\)).
- **Conclusion**: This has a terminating decimal expansion.

### Step 2: Analyze \(\frac{64}{455}\)
- **Denominator**: \(455\)
- **Factorization**:
- \(455 = 5 \times 91\)
- \(91 = 7 \times 13\)
- Therefore, \(455 = 5 \times 7 \times 13\)
- **Form**: The denominator has factors other than \(2\) and \(5\).
- **Conclusion**: This does not have a terminating decimal expansion.

### Step 3: Analyze \(\frac{15}{1600}\)
- **Denominator**: \(1600\)
- **Factorization**:
- \(1600 = 16 \times 100\)
- \(16 = 2^4\) and \(100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2\)
- Therefore, \(1600 = 2^4 \times (2^2 \times 5^2) = 2^6 \times 5^2\)
- **Form**: This is in the form \(2^n \times 5^m\) (where \(n=6\) and \(m=2\)).
- **Conclusion**: This has a terminating decimal expansion.

### Step 4: Analyze \(\frac{13}{3125}\)
- **Denominator**: \(3125\)
- **Factorization**:
- \(3125 = 5^5\)
- **Form**: This is in the form \(5^m\) (where \(m=5\)).
- **Conclusion**: This has a terminating decimal expansion.

### Step 5: Analyze \(\frac{129}{2^2 \times 5^7 \times 7^{17}}\)
- **Denominator**: \(2^2 \times 5^7 \times 7^{17}\)
- **Form**: The denominator has a factor of \(7\), which is not allowed.
- **Conclusion**: This does not have a terminating decimal expansion.

### Step 6: Analyze \(\frac{987}{10500}\)
- **Denominator**: \(10500\)
- **Factorization**:
- \(10500 = 105 \times 100\)
- \(100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2\)
- \(105 = 3 \times 35 = 3 \times 5 \times 7\)
- Therefore, \(10500 = 2^2 \times 5^3 \times 3 \times 7\)
- **Form**: The denominator has factors of \(3\) and \(7\), which are not allowed.
- **Conclusion**: This does not have a terminating decimal expansion.

### Final Conclusion
The rational numbers that have terminating decimal expansions are:
1. \(\frac{17}{8}\)
2. \(\frac{15}{1600}\)
3. \(\frac{13}{3125}\)

Thus, the correct options are 1, 3, and 4.

---