Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?(1 point)
The rational numbers that can be expressed as a repeating decimal are the ones that cannot be simplified to a fraction with a denominator that only contains 2's and 5's as prime factors.
78 = 39 * 2 = 3 * 13 * 2
512 = 256 * 2 = 2^9
1725 = 3^2 * 5^2 * 23
3140 = 2^2 * 5 * 157
Out of these numbers, only 78 can be expressed as a repeating decimal because it is the only one that does not have a denominator with only 2's and 5's as prime factors.
Which of the following rational numbers can be expressed as a repeating decimal: 7/8
, 5/12
, 17/25
, or 31/40
?
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to check if their denominators have any factors other than 2 or 5.
1. For the rational number 78, the numerator and the denominator do not have any common factors other than 2.
2. For the rational number 512, the numerator and the denominator do not have any common factors other than 2.
3. For the rational number 1725, the denominator has a factor of 5.
4. For the rational number 3140, the numerator and the denominator do not have any common factors other than 2.
So, the rational number 1725 can be expressed as a repeating decimal.
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to convert each number into a fraction and then analyze the denominator.
For 78, we can write it as 78/1.
For 512, we can write it as 512/1.
For 1725, we can write it as 1725/1.
For 3140, we can write it as 3140/1.
Now, let's analyze the denominators. If the denominator is divisible only by 2 and/or 5, then the decimal representation will terminate. However, if the denominator has any prime factors other than 2 and 5, then the decimal representation will repeat.
Analyzing the denominators:
78 has a denominator of 1. Since 1 is not divisible by any prime number other than 1, the decimal representation will always terminate.
512 has a denominator of 1. Similar to 78, the decimal representation will terminate.
1725 has a denominator of 1. Again, the decimal representation will terminate.
3140 has a denominator of 1. As before, the decimal representation will terminate.
Therefore, none of the given rational numbers can be expressed as repeating decimals.