To express 711 as a repeating decimal, we start by dividing 1 by 711.
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Perform long division of 1 by 711:
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1.0000000 (we can add decimal places to help with the division)
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711 does not go into 1. It goes into 10 zero times.
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Next, we consider 100 (0.001 in decimal). 711 still does not fit into it. It goes 0 times.
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Now consider 1000. 711 fits into 1000 only once.
\[ 1 \times 711 = 711 \]
Subtract 711 from 1000:
\[ 1000 - 711 = 289 \]
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Bring down another 0 to make 2890.
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711 goes into 2890 four times:
\[ 4 \times 711 = 2844 \]
Subtract 2844 from 2890:
\[ 2890 - 2844 = 46 \]
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Bring down another 0 to make 460.
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711 goes into 460 zero times.
Bring down another 0 to make 4600.
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711 goes into 4600 six times:
\[ 6 \times 711 = 4266 \]
Subtract 4266 from 4600:
\[ 4600 - 4266 = 334 \]
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Bring down another 0 to make 3340.
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711 goes into 3340 four times:
\[ 4 \times 711 = 2844 \]
Subtract 2844 from 3340:
\[ 3340 - 2844 = 496 \]
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Bring down another 0 to make 4960.
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711 goes into 4960 six times:
\[ 6 \times 711 = 4266 \]
Subtract 4266 from 4960:
\[ 4960 - 4266 = 694 \]
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Bring down another 0 to make 6940.
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711 goes into 6940 nine times:
\[ 9 \times 711 = 6399 \]
Subtract 6399 from 6940:
\[ 6940 - 6399 = 541 \]
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Bring down another 0 to make 5410.
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711 goes into 5410 seven times:
\[ 7 \times 711 = 4977 \]
Subtract 4977 from 5410:
\[ 5410 - 4977 = 433 \]
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Bring down another 0 to make 4330.
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711 goes into 4330 six times:
\[ 6 \times 711 = 4266 \]
Subtract 4266 from 4330:
\[ 4330 - 4266 = 64 \]
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Bring down another 0 to make 640.
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711 goes into 640 only 0 times.
Bring down another 0 to make 6400.
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711 goes into 6400 nine times:
\[ 9 \times 711 = 6399 \]
Subtract 6399 from 6400:
\[ 6400 - 6399 = 1 \]
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This gives us a repeating cycle because we've returned to our original value, leading us back to 1.
From the calculations above, the decimal representation of 1/711 is:
\[ 0.001404 \overline{004} \]
Thus, the repeating decimal is:
\[ 0.001404 \overline{004} \]
with 004 repeating.