Which amendment said that states couldn't prevent prevent people from voting based on their race?

I believe the answer is: the 15th amendment

the 15th amendment stated that "right of citizens of the United States to vote shall not be denied or abridged by the United States or by any state on account of race, color, or previous condition of servitude."  The amendemnt was created in response to southern states actions during the reconstruction era which make it extremely hard for African American to vote their representatives in the government.