To express the repeating decimal $0.\overline{56}$ as a ratio of integers, follow these steps:

1. Let $x = 0.\overline{56}$, so $x = 0.56565656\ldots$.
2. Multiply both sides by 100 to shift the decimal two places to the right (since the repeating part is two digits long): $100x = 56.565656\ldots$
3. Now, we have: $100x = 56.565656\ldots \quad \text{and} \quad x = 0.565656\ldots$
4. Subtract the second equation from the first: $100x - x = 56.565656\ldots - 0.565656\ldots$ This simplifies to: $99x = 56$
5. Solve for $x$: $x = \frac{56}{99}$

Thus, $0.\overline{56} = \frac{56}{99}$.

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How do you convert other repeating decimals, like $0.\overline{3}$, into fractions?
2. Why does the method of multiplying by 100 work for two-digit repeating patterns?
3. How can you determine the fraction for decimals with non-repeating and repeating parts, such as $0.123\overline{4}$?
4. Can repeating decimals always be converted into fractions?
5. What is the difference between terminating and repeating decimals in terms of fraction representation?

Tip: When converting repeating decimals to fractions, remember that the number of repeating digits determines the power of 10 used for multiplication.