Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?
A rational number can be expressed as a repeating decimal if and only if it is not a terminating decimal. A terminating decimal is a decimal that ends after a finite number of digits, meaning it does not repeat.
To determine if a rational number is terminating, we need to look at its denominator (the bottom number), which is a power of 5 or 2 when fully simplified.
The prime factorization of 78 is $2\cdot3\cdot13$, so its simplest form is $\frac{13}{2\cdot3}$. Since the denominator contains a factor of 2, 78 cannot be expressed as a repeating decimal.
The prime factorization of 512 is $2^9$, so its simplest form is $\frac{1}{2^6}$. Since the denominator is a power of 2, 512 can be expressed as a terminating decimal.
The prime factorization of 1725 is $3\cdot5^2\cdot23$, so its simplest form is $\frac{23}{3\cdot5^2}$. Since the denominator contains a factor of 5, 1725 cannot be expressed as a repeating decimal.
The prime factorization of 3140 is $2^2\cdot5\cdot157$, so its simplest form is $\frac{157}{2^2\cdot5}$. Since the denominator contains a factor of 5, 3140 cannot be expressed as a repeating decimal.
Out of the given numbers, only 512 can be expressed as a repeating decimal, so the answer is $\boxed{512}$.
Express 713 as a repeating decimal. Mark the repeating digits with a bar.
Note: You will find the option to add a bar above text in the Geometry (∠
) keyboard.
To express 713 as a repeating decimal, we divide 713 by 1, which gives us 713 as the quotient. Since there is no remainder, the decimal representation of 713 is a terminating decimal. Therefore, there are no repeating digits to mark with a bar.
Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?(1 point)
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to examine their decimal representations. A rational number can be expressed as a repeating decimal if and only if it can be written in the form of p/q, where p and q are integers and q is not divisible by any prime number other than 2 or 5.
Let's examine each number:
1. 78: To check if 78 can be expressed as a repeating decimal, we need to divide it by a power of 10. If the division terminates, then it is a terminating decimal, otherwise it repeats. Dividing 78 by 10, we get 7.8, which is a terminating decimal. Therefore, 78 cannot be expressed as a repeating decimal.
2. 512: Dividing 512 by 10, we get 51.2, which is a terminating decimal. So 512 cannot be expressed as a repeating decimal.
3. 1725: Dividing 1725 by 10, we get 172.5, which is a terminating decimal. So 1725 cannot be expressed as a repeating decimal.
4. 3140: Dividing 3140 by 10, we get 314, which is not a terminating decimal. However, dividing 3140 by 100, we get 31.4, which is also not a terminating decimal. Since the division does not terminate with any power of 10, 3140 can be expressed as a repeating decimal.
Therefore, the only rational number that can be expressed as a repeating decimal out of the given options is 3140.
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to check their decimal forms.
A rational number can be expressed as a repeating decimal if and only if its decimal form has a repeating pattern.
Let's examine each of the given numbers:
1. 78: To express 78 as a decimal, we divide 78 by 1. This gives us 78 ÷ 1 = 78. The resulting decimal form, 78.0000..., does not have a repeating pattern. Therefore, 78 cannot be expressed as a repeating decimal.
2. 512: To express 512 as a decimal, we divide 512 by 1. This gives us 512 ÷ 1 = 512. The resulting decimal form, 512.0000..., does not have a repeating pattern. Therefore, 512 cannot be expressed as a repeating decimal.
3. 1725: To express 1725 as a decimal, we divide 1725 by 1. This gives us 1725 ÷ 1 = 1725. The resulting decimal form, 1725.0000..., does not have a repeating pattern. Therefore, 1725 cannot be expressed as a repeating decimal.
4. 3140: To express 3140 as a decimal, we divide 3140 by 1. This gives us 3140 ÷ 1 = 3140. The resulting decimal form, 3140.0000..., does not have a repeating pattern. Therefore, 3140 cannot be expressed as a repeating decimal.
Based on our analysis, none of the given numbers can be expressed as a repeating decimal.