Without actually performing the long division, state whether the rational number will have a terminating decimal expansion or a non - terminating repeating decimal expansion
`(29)/(343)`

Answer

To determine whether the rational number \( \frac{29}{343} \) has a terminating decimal expansion or a non-terminating repeating decimal expansion, we can follow these steps:

### Step 1: Identify the denominator
The rational number given is \( \frac{29}{343} \). Here, the denominator is 343.

### Step 2: Factor the denominator
Next, we need to factor the denominator into its prime factors.

\[ 343 = 7 \times 7 \times 7 = 7^3 \]

### Step 3: Analyze the prime factors
According to the theorem regarding decimal expansions, a rational number \( \frac{p}{q} \) will have a terminating decimal expansion if the denominator \( q \) (in its simplest form) can be expressed as a product of the primes 2 and/or 5 only.

### Step 4: Check the prime factors of the denominator
In our case, the prime factorization of 343 is \( 7^3 \). Since 7 is neither 2 nor 5, we cannot express 343 in the form of \( 2^m \times 5^n \) where \( m \) and \( n \) are non-negative integers.

### Step 5: Conclusion
Since the denominator 343 contains a prime factor (7) that is neither 2 nor 5, the rational number \( \frac{29}{343} \) will have a non-terminating repeating decimal expansion.

Thus, the final answer is:
\[ \text{The rational number } \frac{29}{343} \text{ has a non-terminating repeating decimal expansion.} \]

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