Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?(1 point) EXplain
A rational number can be expressed as a repeating decimal if its decimal representation is non-terminating and there is a repeated pattern.
To determine which of the given numbers can be expressed as a repeating decimal, we need to convert them into decimal form.
1) For 78: The decimal representation is 78.000000... which terminates and does not have a repeating pattern. Therefore, 78 cannot be expressed as a repeating decimal.
2) For 512: The decimal representation is 512.000000... which terminates and does not have a repeating pattern. Therefore, 512 cannot be expressed as a repeating decimal.
3) For 1725: The decimal representation is 1725.000000... which terminates and does not have a repeating pattern. Therefore, 1725 cannot be expressed as a repeating decimal.
4) For 3140: The decimal representation is 3140.000000... which terminates and does not have a repeating pattern. Therefore, 3140 cannot be expressed as a repeating decimal.
None of the given numbers can be expressed as a repeating decimal.
Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?(1 point)
My apologies for the incorrect response in the previous answer.
To determine if a rational number can be expressed as a repeating decimal, we need to check if the number can be written in the form of a fraction where the denominator has prime factors other than 2 or 5.
Let's check each of the given numbers:
1) For 78: If we divide 78 by 1, 2, 3, 6, 13, 26, 39, or 78, we get a whole number. The prime factors of the denominator 78 are 2 and 13, both of which are factors of 10 (2 * 5). Therefore, 78 can be expressed as a terminating decimal (not a repeating decimal).
2) For 512: The prime factors of the denominator 512 are 2 and 2 and 2 and 2 and 2 and 2 and 2 and 2 and 2 (eight 2's), which means the denominator only has the factor 2. Therefore, 512 can be expressed as a terminating decimal.
3) For 1725: The prime factors of the denominator 1725 are 3 and 5 and 5 and 23, which means the denominator has prime factors other than 2 or 5. Therefore, 1725 can be expressed as a repeating decimal.
4) For 3140: The prime factors of the denominator 3140 are 2 and 2 and 5 and 157, which means the denominator has prime factors other than 2 or 5. Therefore, 3140 can be expressed as a repeating decimal.
Therefore, the rational numbers that can be expressed as repeating decimals are 1725 and 3140.