To convert the repeating decimal \( 0.72\overline{72} \) to a rational number, we can follow these steps:
Let \( x = 0.72\overline{72} \).
To eliminate the repeating part, we can multiply \( x \) by a power of 10. Since the repeat is 2 digits long, we'll multiply by \( 100 \):
\[ 100x = 72.\overline{72} \]
Now, we have: \[ x = 0.72\overline{72} \] \[ 100x = 72.\overline{72} \]
Next, we subtract the first equation from the second:
\[ 100x - x = 72.\overline{72} - 0.72\overline{72} \]
This simplifies to:
\[ 99x = 72 \]
Now, divide both sides by \( 99 \):
\[ x = \frac{72}{99} \]
Next, we simplify the fraction. The greatest common divisor (GCD) of 72 and 99 is 9.
Dividing the numerator and the denominator by 9 gives:
\[ x = \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]
Thus, the repeating decimal \( 0.72\overline{72} \) converted to a rational number in simplest form is:
\[ \frac{8}{11} \]