To express \( \frac{711}{999} \) as a repeating decimal, we first divide 711 by 999.
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Perform the Long Division:
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711 divided by 999 is 0 as 711 is less than 999.
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We find \( 7110 \div 999 \). 999 goes into 7110 about 7 times (since \( 999 \times 7 = 6993 \)).
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Subtract \( 6993 \) from \( 7110 \): \[ 7110 - 6993 = 117 \]
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Bring down a 0 to make it \( 1170 \).
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Next, \( 1170 \div 999 \) goes about 1 time (since \( 999 \times 1 = 999 \)).
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Subtract \( 999 \) from \( 1170 \): \[ 1170 - 999 = 171 \]
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Bring down another 0 to make it \( 1710 \).
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Now, \( 1710 \div 999 \) goes about 1 time (again, \( 999 \times 1 = 999 \)).
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Subtract \( 999 \) from \( 1710 \): \[ 1710 - 999 = 711 \]
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Bring down a 0 to make it \( 7110 \).
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At this point, we can see that we have cycled back to \( 7110 \), indicating that the digits we obtained will repeat.
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Express the Result:
The decimal part of our result is \( 0.7111711... \).
Thus, we can write it as:
\[ 0.\overline{711} \]
Where the bar indicates that "711" is the repeating part of the decimal.
So the final answer is:
\[ 0.\overline{711} \]