Question

Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?

Answer 1

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.

Answer 2

Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?(1 point)

Answer 3

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 $\boxed{512}$.

Answer 4

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.