Final answer:
Ivan IV, known as Ivan the Terrible, expanded Russia's territory by annexing the khanates of Kazan, Astrakhan, and Siberia, and by recruiting Cossacks for the colonization of Siberia, making it the most accurate summary of his contribution to Russia's growth as an empire.
Step-by-step explanation:
The best summary of Russia's expansion under Ivan IV during the sixteenth century is that he annexed the khanates of Kazan, Astrakhan, and Siberia, and used the help of Cossacks to colonize Siberia. Thus, of the given options, the most accurate is Ivan IV acquired Siberia with the assistance of Cossacks, rather than despite their refusal. It should be noted that while Ivan IV did take steps to secure access to the Baltic Sea, his central accomplishments for Russian expansion pertain more to his acquisition of territory in the east, such as Siberia, through military conquest.
Ivan IV, also known as Ivan the Terrible, was the first ruler to take on the title of Tsar of all the Russias, a significant move in framing the Russian state as an empire. His military campaigns and colonization efforts significantly transformed Russia into a more powerful nation with expanded borders.