Without actual division, show that each of the rational numbers given below is expressible as a terminating decimal :
`431/500`

Answer

To determine whether the rational number \( \frac{431}{500} \) is expressible as a terminating decimal without performing actual division, we need to analyze the denominator.

### Step-by-Step Solution:

1. **Identify the Rational Number**: The given rational number is \( \frac{431}{500} \).

2. **Examine the Denominator**: We need to check if the denominator (500) can be expressed in the form of \( 2^n \times 5^m \), where \( n \) and \( m \) are non-negative integers.

3. **Factor the Denominator**:
- Start by factoring 500.
- We can divide 500 by 10:
\[ 500 = 50 \times 10 \]
- Factor 50 and 10 further:
\[ 50 = 5 \times 10 = 5 \times (2 \times 5) = 5^2 \times 2 \]
\[ 10 = 2 \times 5 \]
- Now combine these factors:
\[ 500 = 5^2 \times 2 \times 2 \times 5 = 5^3 \times 2^2 \]

4. **Check the Form**: Now we have:
\[ 500 = 2^2 \times 5^3 \]
This confirms that the denominator can be expressed in the required form \( 2^n \times 5^m \).

5. **Conclusion**: Since the denominator \( 500 \) can be expressed as \( 2^2 \times 5^3 \), the rational number \( \frac{431}{500} \) is expressible as a terminating decimal.

### Final Result:
Thus, \( \frac{431}{500} \) is a terminating decimal.

---