Without actually performing the long division, state whether the rational number will have a terminating decimal expansion or a non - terminating repeating decimal expansion
`(64)/(455)`

Answer

To determine whether the rational number \( \frac{64}{455} \) has a terminating decimal expansion or a non-terminating repeating decimal expansion, we can follow these steps:

### Step 1: Identify the denominator
The denominator of the given rational number is \( 455 \).

### Step 2: Prime factorization of the denominator
We need to perform the prime factorization of \( 455 \):
- Start by dividing \( 455 \) by the smallest prime number, which is \( 5 \):
\[ 455 \div 5 = 91 \]
- Next, factor \( 91 \):
- \( 91 \) can be divided by \( 7 \):
\[ 91 \div 7 = 13 \]
- Thus, the complete prime factorization of \( 455 \) is:
\[ 455 = 5 \times 7 \times 13 \]

### Step 3: Analyze the prime factors
For a rational number \( \frac{p}{q} \) to have a terminating decimal expansion, the prime factorization of \( q \) must consist only of the prime factors \( 2 \) and/or \( 5 \).

### Step 4: Check the prime factors of the denominator
From the prime factorization:
\[ 455 = 5 \times 7 \times 13 \]
We can see that besides \( 5 \), there are additional prime factors \( 7 \) and \( 13 \).

### Step 5: Conclusion
Since the denominator \( 455 \) contains prime factors other than \( 2 \) and \( 5 \) (specifically \( 7 \) and \( 13 \)), the rational number \( \frac{64}{455} \) will have a non-terminating repeating decimal expansion.

### Final Answer:
The rational number \( \frac{64}{455} \) has a non-terminating repeating decimal expansion.

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