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 a terminating decimal representation, we need to check if the denominator of each fraction, when expressed in its simplest form, can be written as \(2^m \times 5^n\), where \(m\) and \(n\) are non-negative integers. Let's analyze each case step by step.

### Step-by-Step Solution

1. **For \( \frac{3}{64} \)**:
- The denominator \(64\) can be expressed as \(2^6\) (since \(64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\)).
- Thus, \( \frac{3}{64} = \frac{3}{2^6 \times 5^0} \).
- This is of the form \(2^m \times 5^n\) (where \(m=6\) and \(n=0\)).
- **Conclusion**: This fraction has a terminating decimal representation.

2. **For \( \frac{7}{24} \)**:
- The denominator \(24\) can be expressed as \(2^3 \times 3^1\) (since \(24 = 2 \times 2 \times 2 \times 3\)).
- Thus, \( \frac{7}{24} = \frac{7}{2^3 \times 3^1} \).
- The presence of \(3\) (which is not \(2\) or \(5\)) means it cannot be expressed in the required form.
- **Conclusion**: This fraction does not have a terminating decimal representation.

3. **For \( \frac{17}{400} \)**:
- The denominator \(400\) can be expressed as \(2^4 \times 5^2\) (since \(400 = 2 \times 2 \times 2 \times 2 \times 5 \times 5\)).
- Thus, \( \frac{17}{400} = \frac{17}{2^4 \times 5^2} \).
- This is of the form \(2^m \times 5^n\) (where \(m=4\) and \(n=2\)).
- **Conclusion**: This fraction has a terminating decimal representation.

4. **For \( \frac{1}{1250} \)**:
- The denominator \(1250\) can be expressed as \(2^1 \times 5^4\) (since \(1250 = 2 \times 5 \times 5 \times 5 \times 5\)).
- Thus, \( \frac{1}{1250} = \frac{1}{2^1 \times 5^4} \).
- This is of the form \(2^m \times 5^n\) (where \(m=1\) and \(n=4\)).
- **Conclusion**: This fraction has a terminating decimal representation.

5. **For \( \frac{7}{80} \)**:
- The denominator \(80\) can be expressed as \(2^4 \times 5^1\) (since \(80 = 2 \times 2 \times 2 \times 2 \times 5\)).
- Thus, \( \frac{7}{80} = \frac{7}{2^4 \times 5^1} \).
- This is of the form \(2^m \times 5^n\) (where \(m=4\) and \(n=1\)).
- **Conclusion**: This fraction has a terminating decimal representation.

6. **For \( \frac{21}{500} \)**:
- The denominator \(500\) can be expressed as \(2^2 \times 5^3\) (since \(500 = 2 \times 5 \times 5 \times 5\)).
- Thus, \( \frac{21}{500} = \frac{21}{2^2 \times 5^3} \).
- This is of the form \(2^m \times 5^n\) (where \(m=2\) and \(n=3\)).
- **Conclusion**: This fraction 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**:
- \( \frac{7}{24} \)