Let \( x = 0.\overline{07} \), which means that \( x = 0.070707... \).
To convert this repeating decimal into a fraction, we can eliminate the repeating part by multiplying \( x \) by a power of 10 that will shift the decimal point to the right, just past the repeating section.
Since "07" has two digits, we multiply \( x \) by \( 100 \):
\[ 100x = 7.070707... \]
Now we have two equations:
- \( x = 0.070707... \)
- \( 100x = 7.070707... \)
Next, we subtract the first equation from the second:
\[ 100x - x = 7.070707... - 0.070707... \]
This simplifies to:
\[ 99x = 7 \]
Now, we solve for \( x \):
\[ x = \frac{7}{99} \]
Next, we need to simplify \( \frac{7}{99} \). Since \( 7 \) is a prime number, we check if it divides \( 99 \):
The prime factorization of \( 99 \) is \( 3^2 \times 11 \). Since \( 7 \) does not divide \( 99 \), \( \frac{7}{99} \) is already in its simplest form.
Thus, the repeating decimal \( 0.\overline{07} \) can be expressed as the fraction:
\[ \boxed{\frac{7}{99}} \]