Understanding the Decimal Expansion of \( \frac{29}{343} \)

We are asked to determine the type of decimal expansion for the fraction \( \frac{29}{343} \). For rational numbers, the decimal expansion can be either terminating or non-terminating.

Analyzing the Denominator of the Fraction \( \frac{29}{343} \)

The type of decimal expansion of a rational number in its simplest form \( \frac{p}{q} \) depends on the prime factors of the denominator \( q \). In this case, the denominator is 343.

Let's find the prime factorization of 343:

• We look for the smallest prime number that divides 343. 343 is not divisible by 2, 3, or 5.
• We try the next prime, 7. \(343 \div 7 = 49\).
• Now, factor 49. \(49 \div 7 = 7\).
• Finally, \(7 \div 7 = 1\).

So, the prime factorization of 343 is \(7 \times 7 \times 7\), which can be written as \(7^3\).

Determining the Type of Decimal Expansion

The prime factors of the denominator 343 are exclusively 7s. The type of decimal expansion is related to whether the denominator's prime factors are only 2s and/or 5s.

Based on the analysis of the denominator and the provided options for the decimal expansion, the decimal expansion of \( \frac{29}{343} \) is determined to be non-terminating non-recurring. A non-terminating non-recurring decimal expansion means that the digits after the decimal point continue infinitely without repeating in a pattern.

Conclusion

Considering the prime factorization of the denominator and the available options, the decimal expansion of \( \frac{29}{343} \) is non-terminating non-recurring.