What type of decimal expansion does `(29)/(2^(2) xx 5 xx 7)` have?

Answer

To determine the type of decimal expansion for the fraction \( \frac{29}{2^2 \times 5 \times 7} \), we will analyze the denominator and its factors.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator is \( 2^2 \times 5 \times 7 \).

2. **Factor the Denominator**:
We can break down the denominator:
\[ 2^2 = 4, \quad 5 = 5, \quad 7 = 7 \]
Thus, the denominator can be expressed as:
\[ 4 \times 5 \times 7 \]

3. **Check the Form of the Denominator**:
A fraction has a terminating decimal expansion if its denominator (after simplification) can be expressed in the form \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.

4. **Analyze the Factors**:
In our case, the denominator \( 2^2 \times 5 \times 7 \) contains a factor of \( 7 \). Since \( 7 \) is not of the form \( 2^m \times 5^n \), the denominator does not meet the criteria for a terminating decimal.

5. **Conclusion**:
Since the denominator has a prime factor (7) that is neither 2 nor 5, the decimal expansion of \( \frac{29}{2^2 \times 5 \times 7} \) is a non-terminating repeating decimal.

### Final Answer:
The decimal expansion of \( \frac{29}{2^2 \times 5 \times 7} \) is a **non-terminating repeating decimal**.

---