Why did the shoguns allow the emperor to stay in place?

The emperors have had throughout Japanese history an enormous symbolic power. They were seen as beings of divine origin, living gods, venerated by the people as such. But often, they have not had the real political power, which has belonged to powerful court top officials. However, after a period of long wars among the feudal lords - the daimyos -, Tokugawa Ieyasu managed to get them under his and his clan´s control and set the institution of the shogunate in the early 17th century. The shogun, commander of the army , was the real power. Tokugawa became the shogun and established his seat in Edo (today´s Tokyo). The emperor continued to be a great symbol of national culture and history, a divine being and a link with the ancient past. Such status could not be easily altered or challenged , so it was convenient for shoguns to keep this system of formal, political coexistence with the emperor, as it allowed legitimacy and stability of the social order.

Explanation:

the us constitution is a living document, meaning that it can be changed over time as the country changes.

radioactive dating, also known as  radioisotope dating  is a process that in  estimating the approximate  age  of  rocks. these data can be analyzed from  the studies dealing with the  decay of the radioactive elements that are present in these rocks. in nature, there are a few types of materials or atoms which are quite unstable and they spontaneously decay or breakdown to form other kinds of atoms.