without actual division, find which of the following rational numbers have terminating decimal representation :
`(i)(3)/(64)" "(ii)(7)/(24)" "(iii)(17)/(400)" "(iv)(1)/(1250)" "(vi)(7)/(80)" "(iv)(21)/(500)`

Explanation

To determine which of the given rational numbers have terminating decimal representations, we need to analyze the denominators of each fraction. A rational number in the form \( \frac{p}{q} \) has a terminating decimal representation if the prime factorization of \( q \) (the denominator) contains only the prime factors 2 and/or 5.

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

### Step 1: Analyze \( \frac{3}{64} \)
1. **Factor the denominator**:
\[ 64 = 2^6 \]
2. **Check the prime factors**: The only prime factor is 2.
3. **Conclusion**: Since the denominator has only the prime factor 2, \( \frac{3}{64} \) has a terminating decimal representation.

### Step 2: Analyze \( \frac{7}{24} \)
1. **Factor the denominator**:
\[ 24 = 2^3 \times 3^1 \]
2. **Check the prime factors**: The prime factor 3 is present.
3. **Conclusion**: Since there is a prime factor other than 2 or 5, \( \frac{7}{24} \) does not have a terminating decimal representation.

### Step 3: Analyze \( \frac{17}{400} \)
1. **Factor the denominator**:
\[ 400 = 2^4 \times 5^2 \]
2. **Check the prime factors**: The prime factors are 2 and 5.
3. **Conclusion**: Since the denominator contains only the prime factors 2 and 5, \( \frac{17}{400} \) has a terminating decimal representation.

### Step 4: Analyze \( \frac{1}{1250} \)
1. **Factor the denominator**:
\[ 1250 = 2^1 \times 5^4 \]
2. **Check the prime factors**: The prime factors are 2 and 5.
3. **Conclusion**: Since the denominator contains only the prime factors 2 and 5, \( \frac{1}{1250} \) has a terminating decimal representation.

### Step 5: Analyze \( \frac{7}{80} \)
1. **Factor the denominator**:
\[ 80 = 2^4 \times 5^1 \]
2. **Check the prime factors**: The prime factors are 2 and 5.
3. **Conclusion**: Since the denominator contains only the prime factors 2 and 5, \( \frac{7}{80} \) has a terminating decimal representation.

### Step 6: Analyze \( \frac{21}{500} \)
1. **Factor the denominator**:
\[ 500 = 2^2 \times 5^3 \]
2. **Check the prime factors**: The prime factors are 2 and 5.
3. **Conclusion**: Since the denominator contains only the prime factors 2 and 5, \( \frac{21}{500} \) has a terminating decimal representation.

### Summary of Results
- **Terminating Decimal Representations**:
- \( \frac{3}{64} \)
- \( \frac{17}{400} \)
- \( \frac{1}{1250} \)
- \( \frac{7}{80} \)
- \( \frac{21}{500} \)

- **Non-Terminating Decimal Representation**:
- \( \frac{7}{24} \)