The speed at which the rotating components of a turbine engine spin is not a function of its power, it is a function of the diameter of the rotating assembly.
The larger the diameter, the lower the RPM. The smaller the diameter, the higher the RPM.
For instance, the JT8D turbofan engine has twin rotors, N1 and N2
The N1, which includes the fan, low pressure compressor, and the low pressure turbine has a maximum RPM of about 8,000. The N2
, which includes the high pressure compressor and the high pressure turbine has a maximum RPM of about 12,000.
The RB211-22B's fan rotates at about 3800 RPM. I've forgotten what the max RPM's are for the intermediate and high pressure rotors.
The Allison 501D/T-56 series engines operate at 13,821 RPM. These are single spool engines.
The reason for the speed being tied to the diameter of the rotor is that you don't want your blade tips going faster than around Mach 1; so the bigger diameter rotors spin at lower speeds. Above M=1 and the blades lose efficiency. The same goes for propellers, larger props turn slower.