Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?(1 point)
A rational number can be expressed as a repeating decimal if and only if its denominator contains prime factors other than 2 and 5.
Looking at the given numbers:
78 = (2)(3)(13) contains prime factors other than 2 and 5, so it can be expressed as a repeating decimal.
512 = (2^9) contains only prime factors of 2, so it cannot be expressed as a repeating decimal.
1725 = (3)(5^2)(23) contains prime factors other than 2 and 5, so it can be expressed as a repeating decimal.
3140 = (2^2)(5)(157) contains prime factors other than 2 and 5, so it can be expressed as a repeating decimal.
Therefore, the rational numbers that can be expressed as repeating decimals are 78, 1725, and 3140.
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.
(1 point)
To express 713 as a repeating decimal, we can divide 713 by a power of 10 so that the decimal part becomes a repeating decimal.
Let's divide 713 by 10:
71
______
10 | 713
-70
___
13
Next, we divide 713 by 100:
7.13
________
100 | 713
-700
_____
13
Finally, we divide 713 by 1000:
0.713
_________
1000 | 713
-7000
______
130
-1000
______
300
- 300
______
0
The decimal part does not repeat, so 713 cannot be expressed as a repeating decimal.
wrong
Apologies for the mistake. Let's try to express 713 as a repeating decimal correctly.
To express 713 as a repeating decimal, we can divide 713 by 99 (a two-digit number with both digits being 9). The reason for dividing by 99 is that when dividing a number by 99, the decimal part will repeat.
Let's perform the division:
7.212121...
_________________
99 | 713.000000...
-693
_________________
20
The remainder is 20, and since it is not zero, we can continue the division.
Now, we bring down another zero:
7.212121...
_________________
99 | 713.020000...
-693
_________________
200
- 198
________________
20
The remainder is again 20, and since it is the same as the previous remainder, the pattern will repeat indefinitely.
Therefore, the repeating decimal form of 713 is 7.212121..., with the digits 21 repeating.
To determine which of these rational numbers can be expressed as a repeating decimal, we need to divide each number and check the decimal representation.
1. 78 ÷ 1 = 78. This can be expressed as a terminating decimal because there is no repetition.
2. 512 ÷ 1 = 512. This can also be expressed as a terminating decimal because there is no repetition.
3. 1725 ÷ 1 = 1725. This can be expressed as a terminating decimal because there is no repetition.
4. 3140 ÷ 1 = 3140. This can also be expressed as a terminating decimal because there is no repetition.
Therefore, none of the given numbers 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 check if the numbers satisfy the conditions for a repeating decimal.
A rational number can be expressed as a repeating decimal if and only if its denominator contains factors other than 2 and 5. This is because 2 and 5 are prime factors of the base-10 system, and any fraction with a denominator that only contains these factors will have a terminating decimal representation.
Let's test each of the given numbers:
1. 78: The denominator of 78 is 1 * 78 = 2 * 3 * 13. Since 78 contains factors other than 2 and 5, it can be expressed as a repeating decimal.
2. 512: The denominator of 512 is 1 * 512 = 2^9. Since the denominator only contains the factor 2, it cannot be expressed as a repeating decimal. Instead, it will have a terminating decimal representation.
3. 1725: The denominator of 1725 is 1 * 1725 = 3^3 * 5^2. Since 1725 contains factors other than 2 and 5, it can be expressed as a repeating decimal.
4. 3140: The denominator of 3140 is 1 * 3140 = 2^2 * 5 * 157. Since 3140 contains factors other than 2 and 5, it can be expressed as a repeating decimal.
Therefore, out of the given numbers, 78, 1725, and 3140 can be expressed as repeating decimals.