The decimal expansion of the rational number `(14587)/(1250)` will terminate after:

Options

A. one decimal place
B. two decimal places
C. three decimal places
D. four decimal places

Answer

To determine how many decimal places the decimal expansion of the rational number \( \frac{14587}{1250} \) will terminate after, we need to analyze the denominator and its prime factorization.

### Step-by-Step Solution:

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

2. **Prime factorization of the denominator**:
We need to find the prime factors of \( 1250 \).
- Start by dividing \( 1250 \) by \( 2 \) (the smallest prime number):
\[ 1250 \div 2 = 625 \]
- Next, divide \( 625 \) by \( 5 \):
\[ 625 \div 5 = 125 \]
- Continue dividing \( 125 \) by \( 5 \):
\[ 125 \div 5 = 25 \]
- Divide \( 25 \) by \( 5 \):
\[ 25 \div 5 = 5 \]
- Finally, divide \( 5 \) by \( 5 \):
\[ 5 \div 5 = 1 \]
- The complete prime factorization of \( 1250 \) is:
\[ 1250 = 2^1 \times 5^4 \]

3. **Check the form of the denominator**:
A rational number has a terminating decimal expansion if the prime factorization of its denominator (after simplification) contains only the primes \( 2 \) and \( 5 \).
- Here, \( 1250 = 2^1 \times 5^4 \) contains only \( 2 \) and \( 5 \), so the decimal expansion will terminate.

4. **Determine the number of decimal places**:
To find out how many decimal places the decimal will terminate after, we look at the highest power of \( 10 \) that can be formed from the factors of \( 2 \) and \( 5 \).
- The limiting factor is the smaller exponent between \( 2^1 \) and \( 5^4 \). Here, the exponent of \( 2 \) is \( 1 \).
- Therefore, we can form \( 10^1 \) from \( 2^1 \) and \( 5^1 \), and we have \( 5^3 \) left over.
- The total number of decimal places is determined by the highest power of \( 10 \) that can be formed, which is \( 10^4 \) (since \( 5^4 \) can pair with \( 2^4 \) if we multiply the numerator and denominator by \( 2^3 \)).
- Thus, the decimal will terminate after \( 4 \) decimal places.

### Final Answer:
The decimal expansion of the rational number \( \frac{14587}{1250} \) will terminate after **4 decimal places**.