Without actually performing the long division, state whether the rational number will have a terminating decimal expansion or a non - terminating repeating decimal expansion
`(13)/(3125)`

Answer

To determine whether the rational number \( \frac{13}{3125} \) has a terminating decimal expansion or a non-terminating repeating decimal expansion, we can follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the denominator \( 3125 \) into its prime factors.

\[ 3125 = 5^5 \]

### Step 2: Check the Prime Factors
A rational number has a terminating decimal expansion if the prime factorization of its denominator (after simplification) contains only the prime factors \( 2 \) and/or \( 5 \).

### Step 3: Analyze the Factors
Since the only prime factor of \( 3125 \) is \( 5 \), we can conclude that it meets the criteria for having a terminating decimal expansion.

### Conclusion
Thus, the rational number \( \frac{13}{3125} \) has a terminating decimal expansion.

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