To convert the repeating decimal \( 0.72\overline{72} \) to a rational number, we can set it equal to a variable:
Let \( x = 0.727272\ldots \)
To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating block has two digits):
\[ 100x = 72.727272\ldots \]
Now, we see that the right-hand side can also be expressed in terms of \( x \):
\[ 100x = 72 + x \]
Next, we solve for \( x \) by isolating it:
\[ 100x - x = 72 \] \[ 99x = 72 \] \[ x = \frac{72}{99} \]
Now, we simplify \( \frac{72}{99} \). To do this, we find the greatest common divisor (GCD) of 72 and 99.
The prime factorization of 72 is \( 2^3 \times 3^2 \), and the prime factorization of 99 is \( 3^2 \times 11 \). The GCD is \( 3^2 = 9 \).
Now, we simplify:
\[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]
Thus, the repeating decimal \( 0.72\overline{72} \) as a rational number in simplest form is:
\[ \boxed{\frac{8}{11}} \]