[tex]x=0.5777...\ \ \ |multiply\ both\ sides\ by\ 10\\\\10x=5.777...\ \ \ \ |multiply\ both\ sides\ by\ 10\\\\100x=57.777...\\\\make\ a\ difference\ 100x-10x:\\\\90x=57.777...-5.777...\\\\90x=52\ \ \ \ |divide\ both\ sides\ by\ 90\\\\x=\frac{52}{90}\\\\x=\frac{26}{45}\\\\\boxed{0.5777...=\frac{26}{45}}[/tex]
