Without actual division, show that each of the rational numbers given below is expressible as a terminating decimal :
`807/1250`

Answer

To determine whether the rational number \( \frac{807}{1250} \) is expressible as a terminating decimal without performing actual division, we need to analyze the denominator after simplifying it into its prime factors. A rational number can be expressed as a terminating decimal if its denominator, when in simplest form, consists only of the prime factors 2 and/or 5.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the given rational number is \( 1250 \).

2. **Factor the Denominator**:
We need to factor \( 1250 \) into its prime factors.
- Start by dividing \( 1250 \) by \( 2 \) (the smallest prime number):
\[ 1250 \div 2 = 625 \]
- Now, factor \( 625 \):
\[ 625 \div 5 = 125 \quad (\text{since } 625 \text{ is not divisible by } 2) \]
\[ 125 \div 5 = 25 \]
\[ 25 \div 5 = 5 \]
\[ 5 \div 5 = 1 \]
- Thus, we can express \( 1250 \) as:
\[ 1250 = 2^1 \times 5^4 \]

3. **Check the Prime Factors**:
The prime factorization of \( 1250 \) is \( 2^1 \times 5^4 \). Since the only prime factors are \( 2 \) and \( 5 \), we can conclude that the denominator does not contain any other prime factors.

4. **Conclusion**:
Since the denominator \( 1250 \) can be expressed as \( 2^1 \times 5^4 \), the rational number \( \frac{807}{1250} \) is expressible as a terminating decimal.

### Final Answer:
Therefore, \( \frac{807}{1250} \) is a terminating decimal.

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