Question

Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point) $$

Answer 1

To express \( \frac{8}{11} \) as a repeating decimal, we can perform the long division of 8 by 11.

1. Divide 8 by 11. Since 8 is less than 11, we place a decimal point and add a zero. This allows us to divide 80 by 11.

2. \( 11 \) goes into \( 80 \) a total of \( 7 \) times (since \( 11 \times 7 = 77 \)).

3. Subtract \( 77 \) from \( 80 \) to get \( 3 \).

4. Bring down another \( 0 \), making it \( 30 \).

5. \( 11 \) goes into \( 30 \) \( 2 \) times (since \( 11 \times 2 = 22 \)).

6. Subtract \( 22 \) from \( 30 \) to get \( 8 \).

7. Bring down another \( 0 \), making it \( 80 \) again.

8. At this point, we see that we have returned to \( 80 \), which means the sequence will repeat.

Thus, the decimal representation of \( \frac{8}{11} \) is:

\[
0.\overline{72}
\]

The repeating decimal is \( 0.\overline{72} \), with the digits \( 72 \) repeating indefinitely.