1. The problem is to express the repeating decimal $0.5\dot{8}\dot{8}$ as a fraction in simplest form. 2. Let $x = 0.5\dot{8}\dot{8}$, which means $x = 0.588888\ldots$ where the digit 8 repeats infinitely. 3. Multiply $x$ by 10 to shift the decimal point one place: $$10x = 5.88888\ldots$$ 4. Multiply $x$ by 100 to shift the decimal point two places: $$100x = 58.88888\ldots$$ 5. Subtract the equation from step 3 from the equation in step 4 to eliminate the repeating part: $$100x - 10x = 58.88888\ldots - 5.88888\ldots$$ $$90x = 53$$ 6. Solve for $x$: $$x = \frac{53}{90}$$ 7. Check if the fraction can be simplified. The numerator 53 is prime and does not share factors with 90, so the fraction is already in simplest form. Final answer: $$0.5\dot{8}\dot{8} = \frac{53}{90}$$