What power is exercised when state legislatures or national conventions vote for or against a constitutional amendment?

The power exercised is popular sovereignty.

Explanation:

Popular sovereignty is the doctrine by which the state is created and subject to the will of the people, who are the source of all political power. This doctrine occurs when state legislatures or national conventions vote for or against a constitutional amendment and is closely associated with contractualist philosophers, among them Thomas Hobbes, John Locke, Jean-Jacques Rousseau, Voltaire and Baron de Montesquieu.

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