The word gangster refers to a group of vicious criminals who usually operate in an organized manner.
The word Axis Powers here refers to the coalition of Germany, Italy, and Japan. So, because the attack was jointly carried out by this group of states, Roosevelt in an attempt to brand them accordingly used the word gangsters.
2. The general context and language of the speech in the passage serves to underscore the criminality of the actions of the Axis Powers.
In the context in which Roosevelt used those words, he was referring to the above-named coalition. His spefdech was a call to all countries of the world to bear America witness against the actions of the Axis Powers and in a way, to absolve the United States of America from the consequences of any retaliatory actions that she was prepared to carry out against the attackers.
History shows that the US's retaliation was devastating and lasted generations afterward.
Roosevelt was not only incensed about the attack on America but delivered the above speech using words such as gangster to connote that their actions were criminal and unpopular. One could say that his objective was to ensure that listeners would become sympathetically inflamed as well.
The central idea of Roosevelt's delivery, therefore, is that no country has the right to deny other states the entitlement to live in harmony with its neighbors and that to attack a neighbor was a criminal action.
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