From wikipedia: "The royal prerogative includes the powers to appoint and dismiss ministers, regulate the civil service, issue passports, declare war, make peace, direct the actions of the military, and negotiate and ratify treaties, alliances, and international agreements."
This isn't as black and white as you are proposing, there is nuance.
> it's not like the king of england has any real power
This is a widely-held belief.
The crown rarely if ever exerts such powers, and on that rare occasion it does so on the advice of the parliment.
Technically such a power exists, but calling it "real" is silly.
Look at the countries of the Commonwealth. The crown appoints a governor general based on the recommendation of the prime minister. This person acts in a ceremonial capacity, and on behalf of the crown. The handful of times a GG has actually acted have been within the interests of the country, have had little to no input from the crown, and are decades apart.