Without doing any actual division, find which of the following rational numbers have terminating decimal representation :
(i) `7/16" (ii) "23/125" (iii) "9/14`

Answer

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

Let's analyze each of the given fractions step by step:

### Step 1: Analyze \( \frac{7}{16} \)
1. **Identify the denominator**: The denominator is \(16\).
2. **Factor the denominator**: \(16 = 2^4\).
3. **Check the form**: We can express \(16\) as \(2^4 \times 5^0\) (since \(5^0 = 1\)).
4. **Conclusion**: Since \(16\) can be expressed in the required form, \( \frac{7}{16} \) has a terminating decimal representation.

### Step 2: Analyze \( \frac{23}{125} \)
1. **Identify the denominator**: The denominator is \(125\).
2. **Factor the denominator**: \(125 = 5^3\).
3. **Check the form**: We can express \(125\) as \(2^0 \times 5^3\).
4. **Conclusion**: Since \(125\) can be expressed in the required form, \( \frac{23}{125} \) has a terminating decimal representation.

### Step 3: Analyze \( \frac{9}{14} \)
1. **Identify the denominator**: The denominator is \(14\).
2. **Factor the denominator**: \(14 = 2^1 \times 7^1\).
3. **Check the form**: The denominator contains a factor of \(7\), which cannot be expressed in the form \(2^m \times 5^n\).
4. **Conclusion**: Since \(14\) cannot be expressed in the required form, \( \frac{9}{14} \) does not have a terminating decimal representation.

### Final Conclusion:
- \( \frac{7}{16} \) has a terminating decimal representation.
- \( \frac{23}{125} \) has a terminating decimal representation.
- \( \frac{9}{14} \) does not have a terminating decimal representation.