in which domain should a biologist classify a single-celled organism containing a nucleus
The single-celled organism containing a nucleus should be classified in the domain called Eukarya.
To arrive at this classification, scientists use a systematic approach called taxonomy. Taxonomy classifies living organisms into different categories called taxa. The highest level of classification is the domain. There are three domains: Bacteria, Archaea, and Eukarya.
Organisms in the domain Bacteria are prokaryotes, meaning they lack a nucleus. Organisms in the domain Archaea are also prokaryotes but have distinct characteristics that differentiate them from bacteria.
On the other hand, organisms in the domain Eukarya are eukaryotes, meaning they have a true nucleus, which contains their genetic material. Eukaryotes include various organisms such as animals, plants, fungi, and protists.
Since the single-celled organism you mentioned contains a nucleus, it falls into the domain Eukarya.