Let \( x = 0.72\overline{72} \). This means that \( x = 0.72727272\ldots \).
To eliminate the repeating part, multiply \( x \) by 100 (since the repeating part has 2 digits):
\[ 100x = 72.727272\ldots \]
Now, we have two equations:
\[ x = 0.727272\ldots \quad (1) \]
\[ 100x = 72.727272\ldots \quad (2) \]
Next, we subtract equation (1) from equation (2):
\[ 100x - x = 72.727272\ldots - 0.727272\ldots \]
This simplifies to:
\[ 99x = 72 \]
Now, we can solve for \( x \):
\[ x = \frac{72}{99} \]
Next, we simplify the fraction \( \frac{72}{99} \). First, we find the greatest common divisor (GCD) of 72 and 99.
The prime factorization of 72 is \( 2^3 \times 3^2 \) and for 99, it is \( 3^2 \times 11 \). The GCD is \( 3^2 = 9 \).
Now we divide both the numerator and denominator by 9:
\[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]
Thus, the simplest form of \( 0.72\overline{72} \) as a rational number is
\[ \boxed{\frac{8}{11}} \]
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