To determine which of the given rational numbers can be expressed as a repeating decimal, we need to look at their fractional or decimal representations.
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78%: As a fraction, \( \frac{78}{100} \) simplifies to \( \frac{39}{50} \). Since 50 has only 2 in its prime factorization, this will be a terminating decimal.
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512: This is an integer and can be written as \( \frac{512}{1} \). Integers are also considered terminating decimals.
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1725: Similar to 512, this is also an integer ( \( \frac{1725}{1} \) ) and is therefore a terminating decimal.
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3140: Again, this is an integer ( \( \frac{3140}{1} \) ), which means it's a terminating decimal too.
Since all the numbers listed are either terminating decimals or integers, none of them can be expressed as a repeating decimal.
For the second part, let's express \( \frac{713}{13} \) as a decimal and check if it repeats:
- Dividing 713 by 13:
- \( 13 \) into \( 71 \) goes \( 5 \) times (because \( 13 \times 5 = 65 \)).
- Subtract \( 65 \) from \( 71 \) to get \( 6 \).
- Bring down the next digit (which is 3) to get \( 63 \).
- \( 13 \) into \( 63 \) goes \( 4 \) times (because \( 13 \times 4 = 52 \)).
- Subtract \( 52 \) from \( 63 \) to get \( 11 \).
- Bring down a zero (from decimal) to get \( 110 \).
- \( 13 \) into \( 110 \) goes \( 8 \) times (because \( 13 \times 8 = 104 \)).
- Subtract \( 104 \) from \( 110 \) to get \( 6 \).
- Again bring down a zero to get \( 60 \).
- \( 13 \) into \( 60 \) goes \( 4 \) times (because \( 13 \times 4 = 52 \)).
- Subtract \( 52 \) from \( 60 \) to get \( 8 \).
- Bring down a zero to get \( 80 \).
- \( 13 \) into \( 80 \) goes \( 6 \) (because \( 13 \times 6 = 78 \)).
- Subtract \( 78 \) from \( 80 \) to get \( 2 \).
- Bring down a zero to get \( 20 \).
- \( 13 \) into \( 20 \) goes \( 1 \) time ( \( 13 \times 1 = 13 \)).
- Subtract \( 13 \) from \( 20 \) to get \( 7 \).
- Bring down a zero to get \( 70 \).
- \( 13 \) into \( 70 \) goes \( 5 \) times ( \( 13 \times 5 = 65 \)).
- Subtract \( 65 \) from \( 70 \) to get \( 5 \).
- Bring down a zero to get \( 50 \).
- \( 13 \) into \( 50 \) goes \( 3 \) times \( (39) \).
- Subtract \( 39 \) from \( 50 \) to get \( 11 \), and the cycle continues.
From this division, we can see that it will repeat this process because it returns to the remainder of \( 11 \).
So, \( \frac{713}{13} \) can be expressed as:
\[ 54.230769230769... \]
Thus, we can denote the repeating part. The repeating block here is \( 230769 \).
Hence, expressed with a repeating bar, it is:
\[ 54.\overline{230769} \]