To express \( \frac{7}{13} \) as a repeating decimal, we can perform long division.
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Start by dividing 7 by 13:
- 13 goes into 70 five times (since \( 13 \times 5 = 65 \)).
- Subtract \( 65 \) from \( 70 \) to get \( 5 \).
- Bring down a zero to make it \( 50 \).
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Now divide \( 50 \) by \( 13 \):
- 13 goes into 50 three times (since \( 13 \times 3 = 39 \)).
- Subtract \( 39 \) from \( 50 \) to get \( 11 \).
- Bring down a zero to make it \( 110 \).
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Now divide \( 110 \) by \( 13 \):
- 13 goes into 110 eight times (since \( 13 \times 8 = 104 \)).
- Subtract \( 104 \) from \( 110 \) to get \( 6 \).
- Bring down a zero to make it \( 60 \).
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Now divide \( 60 \) by \( 13 \):
- 13 goes into 60 four times (since \( 13 \times 4 = 52 \)).
- Subtract \( 52 \) from \( 60 \) to get \( 8 \).
- Bring down a zero to make it \( 80 \).
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Now divide \( 80 \) by \( 13 \):
- 13 goes into 80 six times (since \( 13 \times 6 = 78 \)).
- Subtract \( 78 \) from \( 80 \) to get \( 2 \).
- Bring down a zero to make it \( 20 \).
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Now divide \( 20 \) by \( 13 \):
- 13 goes into 20 one time (since \( 13 \times 1 = 13 \)).
- Subtract \( 13 \) from \( 20 \) to get \( 7 \).
- Bring down a zero to make it \( 70 \).
At this point, we notice that we are back at the beginning of the cycle where we started with \( 70 \). This means the decimal repeats.
Thus, the decimal representation of \( \frac{7}{13} \) is:
\[ 0.\overline{538461} \]
So, \( \frac{7}{13} = 0.\overline{538461} \).