State Department spokeswoman Victoria Nuland added that North Korea's test was unrelated to the president's position that the U.S. has "more nuclear weapons than we need."
"We're obviously talking about apples and oranges in the context of the U.S. having almost 1,700 nuclear weapons and we are working to prevent (North Korea) from having the effective ability to launch a far smaller amount," she said.
Defense Secretary Leon Panetta, in remarks Tuesday morning to Pentagon workers gathered in the building's courtyard, said the U.S. is going to have to continue to deal with rogue states like North Korea.
"We just saw what North Korea has done in these last few weeks, a missile test and now a nuclear test," he said. "They represent a serious threat to the United States of America, and we've got to be prepared to deal with that. "
The chairman of the House Foreign Affairs Committee, California Republican Ed Royce, called for even tougher sanctions and demanded that the administration "replace its failed North Korea policy."
"Otherwise, the grave North Korean threat to the region and the United States will only grow," Royce said.
And House Intelligence Committee chairman Mike Rogers, R-Mich., added that the U.S. must take a new approach to dealing with North Korea. "The key to stemming North Korea's cycle of provocation is to seriously engage the Chinese in exercising leverage over their neighbor," he said.
On Tuesday, China expressed firm opposition to the test but called for a calm response by all sides.
Associated Press writer Lolita C. Baldor and AP Intelligence Writer Kimberly Dozier contributed to this report.
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