The correct answer is the Chalukyas of Badami.
The Chalukyas of Badami were a powerful empire that ruled over much of southern India from the 6th to the 8th centuries CE. They were founded by Pulakeshin I, who conquered the Kadambas of Goa in 540 CE. This defeat led to the decline of the Kadambas and the rise of the Chalukyas as the dominant power in southern India.
The Chalukyas were a Hindu empire, and their rulers were patrons of the arts and architecture. They built many temples and other monuments, including the famous rock-cut temples at Badami, Aihole, and Pattadakal.
The Chalukyas were eventually overthrown by the Rashtrakutas in the 8th century CE. However, they left a lasting legacy on Indian history and culture.
The other options are incorrect because they did not defeat the Kadambas of Goa. The Pallavas of Kanchi were a rival empire to the Chalukyas, and they fought several wars against each other. However, the Pallavas never conquered the Kadambas. The Rashtrakutas of Manyakheta were another rival empire to the Chalukyas, and they eventually defeated them in the 8th century CE. However, they did not defeat the Kadambas before that time. The Cholas of Thanjavur were a powerful empire that ruled over much of southern India from the 9th to the 13th centuries CE. However, they did not come into existence until after the decline of the Kadambas.