DENVER » Buoyed by a powerful debate showing, Mitt Romney said today he offers “prosperity that comes through freedom” to a country struggling to shed a weak economy. President Barack Obama accused the former Massachusetts governor of running from his own record in pursuit of political power.

Both men unleashed new attack ads in the battleground states in a race with little more than a month to run, Obama suggesting Romney couldn’t be trusted with the presidency, and the Republican accusing the president of backing a large tax increase on the middle class.

Not even Democrats disputed that Romney was likely to benefit politically from the debate Wednesday night in which he aggressively challenged Obama’s stewardship of the economy and said his own plans would help pull the country out of a slow-growth rut. Still, there was no immediate indication that the race would expand beyond the nine battleground states where the rivals and their running mates spend nearly all of their campaign time and advertising dollars.

Debate host Colorado is one of them, and Virginia, where Romney headed for an evening speech, is another. So, too, Wisconsin, Obama’s destination for a mid-day rally. Nevada, Ohio, Iowa, New Hampshire, Florida and North Carolina are the others.

Among them, the nine states account for 110 electoral votes out of the 270 needed to win the White House, more than enough to tip the campaign to one man or the other.

“Victory is in sight,” Romney exulted in an emailed request for donations to supporters. It was a show of confidence by a man hoping for a quick reversal in pre-debate public opinion polls that showed him trailing in battleground states as well as nationally.

Reprising a line from the debate, he told an audience of conservatives in Denver that Obama offers “trickle-down government.” He added, “I don’t think that’s what America believes in. I see instead a prosperity that comes through freedom.”

Another possible pivot point in the campaign neared in the form of Friday’s government report on unemployment for September. Joblessness was measured at 8.1. percent the previous month.

Obama campaigned with the energy of a man determined to make up for a subpar debate showing. Speaking to a crowd not far from the debate hall, he said mockingly that a “very spirited fellow” who stood next to him onstage Wednesday night “does not want to be held accountable for the real Mitt Romney’s positions” on taxes, education and other issues. “Governor Romney may dance around his positions, but if you want to be president you owe the American people the truth,” he said.

Later, before a crowd of tens of thousands in Madison, Wis., he said Romney wants to cut federal funding for Public Television while repealing legislation that regulates the banking industry “I just want to make sure I’ve got this straight: He’ll get rid of regulations on Wall Street, but he’s going to crack down on Sesame Street,” Obama said.

Taxes were a particular point of contention between the two men, although they were sharply divided as well on steps the cut the deficit, on government regulation, on education and Medicare.

Both in the debate and on the day after, Obama said repeatedly that his rival favors a $5 trillion tax cut that is tilted to the wealthy and would mean tax increases on the middle class or else result in a spike in federal deficits.

Romney said it wasn’t so, and counterattacked in a new television commercial. It cited a report by the American Enterprise Institute that said Obama and “his liberal allies” want to raise taxes on middle class earners by $4,000 and that the Republican alternative would not raise the amount they owe to the IRS.

Romney has refused so far to disclose many of the details to support his assertion that his proposal would not lead to a tax cut. His ad was an attempt to parry a report by the Tax Policy Center that Obama has frequently tried used to political advantage, as he did again during the day.

In a new ad by the president’s campaign, Romney is quoted as saying that a $5 trillion tax cut “is not my plan.” The ad then cites a study by the Tax Policy Center as saying it is, and asks why the Republican challenger “won’t level with us about his tax plan which gives the wealthy huge new tax breaks.

“Because if we can’t trust him here” — a photo of the debate stage appears — “How could we ever trust him here,” the narrator says as a photo of the Oval Office fills the screen.

The two men debate twice more this month, Oct. 16 in Hempstead, N.Y. and Oct. 22 in Boca Raton, Fla.

Before they do, Vice President Joe Biden and Romney’s running mate, Wisconsin Rep. Paul Ryan, will share a stage in Danville, Ky. in one week’s time.

Biden plunged into the tax debate during the day, saying the administration does indeed want to increase the taxes paid by the wealthy by $1 trillion.

“We want to let that trillion-dollar tax cut expire so the middle class doesn’t have to bear the burden of all that money going to the super wealthy,” he said while campaigning in Iowa. “That’s not a tax raise, that’s called fairness where I come from.”

Republicans didn’t see it that way, and seized on the comment as evidence the administration’s policies would kill jobs.

Whatever the eventual outcome of the race, Romney seemed to have achieved his goal of a campaign reset. Democrats braced for tightening polls over the next several days in the states where the campaign will be won or lost.

The head of one Republican-aligned independent group said all such organizations should consider expanding into states that have effectively been written off. “If we didn’t get a home run, we certainly got a triple” from Romney’s showing in the debate, said American Future Fund’s founder Nick Ryan, who sided with Rick Santorum during the primaries.

Obama campaign strategist David Axelrod acknowledged in a conference call with reporters that an adjustment in strategy would be needed in the debates to come. “Obviously moving forward we’re going to take a hard look at this, and we’re going to have to make some judgments as to where to draw the line in these debates and how to use our time,” Axelrod said.

Romney frequently interrupted both Obama and moderator Jim Lehrer of the Public Broadcasting Service during the 90-minute debate, sometimes talking over one or both of them to argue that the president’s policies hadn’t restored the economy, or alternatively, that the president was making false accusations about Republican proposals.

While both men prepared extensively for their first head-to-head encounter, Romney had the advantage of having taken part in 19 debates with his Republican rivals over the course of many months. He seemed to employ many of the techniques that he honed then, insisting on speaking time he claimed he was entitled to, for example, generally without seeming belligerent.

The president’s last prior debate was four years ago, when he was running against Sen. John McCain.

Associated Press writers Philip Elliott and Beth Fouhy in Washington, Julie Pace in Madison, Wis., and Matthew Daly in Council Bluffs, Iowa. contributed to this story. Espo reported from Washington.