What kind of power is the power to investigate?

Investigative power is judical power, if done by a legislative body (such as a committee on Governmental Oversite - e.g., Watergate, or corruption hearings on elected officials by a State Legislature, etc.) because potential legal charges could result.

If the investigation is being done by law enforcement, it is a legal power -again- because a potential crime has been assumed and potential legal charges (amounting to misdemeanor or felony level, if criminal, or warning, or administrative charges, if not of the scale of misdemeanors, could result.

If done by journalists in the process of investigating facts and accusations for a story it could be non-legal powers assumed by the '4th Estate' as part of their job to oversee government, root out corruption, get the facts and the basis for the full story (both sides!) to make an objective and complete examination of relevant facts.

Then there is the curious potential of an individual investigating the background of a person they will be dating, want to date or one who is dating an ex-spouse or daughter/son or even grand-daughter/grandson. That 'power' is pure unadulterated curiOSity, no legal basis at all. Done for the best of intentions, perhaps: for the safety of a loved one - they tell themselves - or for the protection of financial assets, or protection from potential 'gold-diggers' or such rational-izations, they convince themselves. But, as the fable says, the power of curiOSity has a 'dark-side' which could 'kill the cat;' and, also as the saying goes, 'satisfaction brought it back." You choose.