The Dow ended half a percent higher. The S&P 500 finished up seven tenths of a percent, while the Nasdaq rose a full percent.
Powell said at a forum sponsored by the Swedish central bank that the Fed's independence is essential for it to battle inflation, but he offered no clues on future interest rate hikes.
George Ball is chairman of Sanders Morris Harris.
"He was quite careful not to say anything that would tip his hand or tip his hand differently as to the monetary policy so that there was no new light shed on what the Fed may do next. And that's really the big question right now. Everybody wants direction from the Fed, and that direction may be a year or a year and a half off... Right now, the Fed is going to stay hawkish in tone. They are going to stay even handed and will raise rates in more moderate steps, but raise rates to something like a 5% or 5.25% level over the first half of the year. And then we'll probably just keep it there for a long time until circumstances change substantially."
Tech shares, including Amazon and Microsoft, gave the S&P 500 its biggest boost on Tuesday.
Shares of Broadcom fell, but pared much of the losses by the close, after a report said Apple plans to replace a Broadcom chip from its devices with an in-house design in 2025.
Shares of investment bank Jefferies Financial rose, a day after it posted its second-best year for investment banking revenue.
And shares of Bed Bath & Beyond jumped more than 27% as retail investors speculated it could be a potential acquisition target and as short-sellers closed out their bets on the stock falling.
The struggling retailer on Tuesday also said it will lay off more employees in an attempt to reduce costs, a week after announcing it was considering filing for bankruptcy.
