Which of the following rational numbers can be expressed as a repeating decimal: 78 , 512 , 1725 , or 3140 ?(1 point)
In order for a rational number to be expressed as a repeating decimal, its decimal representation must have a repeating pattern.
To determine if a rational number has a repeating decimal pattern, we can consider its prime factorization. If the prime factorization of the denominator contains any prime factors other than 2 or 5, then the decimal representation will repeat.
Let's look at the prime factorizations of the given numbers:
78 = 2 * 3 * 13
512 = 2^9
1725 = 3 * 5^2 * 23
3140 = 2^2 * 5 * 157
From the prime factorizations, we can see that none of the numbers have denominators with prime factors other than 2 or 5. Therefore, none of the given rational numbers can be expressed as a repeating decimal.
Answer: None of these rational numbers can be expressed as a repeating decimal.