WASHINGTON — The gap between the wealthy and the poor is most extreme in several of the country’s largest and most prosperous cities, according to a Brookings Institution study released Thursday.The economic divides in Atlanta, San Francisco, Washington, New York, Chicago and Los Angeles are significantly greater than in cities in the rest of the country, according to the study from the Washington-based think tank.Among the 50 biggest cities, Fort Worth and Arlington have smaller income gaps. Incomes for the top 5 percent of earners in Atlanta, which has the biggest income gap, averaged $279,827 in 2012. That’s almost 19 times as much as what the bottom 20 percent of the city’s population earned. The ratio is more than double the nationwide average for this measure of income inequality. The top 5 percent of earners nationwide have incomes 9.1 times as great as the bottom quintile. In Arlington, which ranked 49th for inequality, the top 5 percent of residents earn an average of $175,759, or 7.3 times as much as the bottom 20 percent. Fort Worth ranked No. 44, with the top 5 percent earning $168,989, or 8.1 times as much as the bottom 20 percent. In Dallas, the gap is wider, with the top 5 percent averaging $200,367, 11.2 times as much as the bottom 20 percent. That put Dallas at No. 13 among U.S. cities.Jeff Ferrell, a TCU sociology professor, said Fort Worth has worked hard to keep its older neighborhoods “vibrant,” which may have helped prevent the wealth gap from widening.“I’m not an economist, but I do know, in general, that when you have neighborhoods with a more integrated urban core, it tends to mean you’re not going to have as big a gap from one area to the next,” Ferrell said.Cities with the biggest income gaps may struggle to provide adequate public schooling and municipal services because of a narrow tax base, and they “may fail to produce housing and neighborhoods accessible to middle-class workers and families,” the study said. The findings come at a delicate moment for the country, still slogging through a weak recovery from the Great Recession. Much of the nation’s job growth has been concentrated in lower-wage careers. Few Americans have enjoyed pay raises. President Barack Obama is pushing for a higher minimum wage.Major chasms have appeared in the tech hub of San Francisco, the financial center of New York, the seat of the federal government in Washington and the home of the entertainment industry in Los Angeles. “In San Francisco, skyrocketing housing costs may increasingly preclude low-income residents from living in the city altogether,” the study said. Not all tech hubs have witnessed rising inequality. In Seattle, where Amazon and Microsoft are based, income disparity has declined since 2007. Austin experienced a mild uptick.Of the 50 biggest cities, just 18 have experienced greater income inequality since the recession. That was due primarily to falling incomes for the poorest residents. That occurred in places that suffered when the housing bubble burst — such as Tucson, Ariz., and Albuquerque — and Midwestern cities still reeling from the collapse of manufacturing, such as Cleveland, Indianapolis and Milwaukee. Staff writers Steve Kaskovich and Bill Miller contributed to this report.