Let \( x = 0.72\overline{72} \), which means the decimal repeats indefinitely as \( 0.727272...\).
To eliminate the repeating decimal, we can express \( x \) in terms of an equation. First, we multiply \( x \) by 100 (since the repeating part has two digits):
\[ 100x = 72.727272... \]
Now we write the original \( x \) under this new equation:
\[ x = 0.727272... \]
Next, we will subtract the second equation from the first:
\[ 100x - x = 72.727272... - 0.727272... \]
This gives us:
\[ 99x = 72 \]
Now, we solve for \( x \):
\[ x = \frac{72}{99} \]
Next, we simplify this fraction. To do so, we find the greatest common divisor (GCD) of 72 and 99. The prime factorization of each number is:
- \( 72 = 2^3 \times 3^2 \)
- \( 99 = 3^2 \times 11 \)
The GCD is \( 9 \) (since \( 3^2 \) is the common factor).
Now we divide both the numerator and the denominator by the GCD:
\[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]
Therefore, the rational number in simplest form that corresponds to \( 0.72\overline{72} \) is:
\[ \boxed{\frac{8}{11}} \]
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