State whether `(6)/(200)` has terminating or non-terminating repeating decimal expansion.

Answer

To determine whether the fraction \( \frac{6}{200} \) has a terminating or non-terminating repeating decimal expansion, we can follow these steps:

### Step 1: Simplify the Fraction
First, we simplify the fraction \( \frac{6}{200} \).

\[ \frac{6}{200} = \frac{6 \div 2}{200 \div 2} = \frac{3}{100} \]

### Step 2: Prime Factorization of the Denominator
Next, we need to perform the prime factorization of the denominator, which is 100.

\[ 100 = 10 \times 10 = (2 \times 5) \times (2 \times 5) = 2^2 \times 5^2 \]

### Step 3: Check the Form of the Denominator
A fraction has a terminating decimal expansion if the prime factorization of its denominator (after simplification) consists only of the primes 2 and 5.

From our factorization, we see:

\[ 100 = 2^2 \times 5^2 \]

This confirms that the denominator is of the form \( 2^m \times 5^n \).

### Step 4: Conclusion
Since the denominator of the simplified fraction \( \frac{3}{100} \) is of the form \( 2^m \times 5^n \), we conclude that \( \frac{6}{200} \) has a terminating decimal expansion.

### Final Answer
Thus, \( \frac{6}{200} \) has a terminating decimal expansion.

\[ \text{Decimal representation: } \frac{3}{100} = 0.03 \]

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