TV Tropes claims Super Intelligence breaks down into 4 main tropes (their names, my summary):
- Super learning and eidetic memory are the ability to 'master' a new subject quickly.
- Advanced reasoning is the ability to diagnose problems and come up with novel solutions, presumably from their encyclopedic knowledge of science, history, geography, trivia....
- Exceptional Perception is the ability to perceive details and extrapolate their significance within the context.
- Manipulator Extraordinaire is the ability to predict other people's behavior, presumably through theory of mind and probability analysis.
They also acknowledge that just being a general science nerd seems to be a (mostly useless?) part of the trope, and they also have a page for Intelligence Tropes which might need to be avoided. They are not a writer's resource, just a list of pop culture. It's not going to help you make characters.
Assuming your super-intelligent AI are created for a purpose (not born haphazardly), I can think of corporate entities that might 'invest' in their creation. You'll need to figure out how they became emancipated, if they were. They may have contracts like indentured servants, with a retirement plan as they age out of the workforce. They way things look now that might be a cycle of 5-10 years, like serving in the military or buying a desktop.
Consider how nations today invest in education initiatives that will pay off decades later, or building infrastructure as a longterm investment in future generations. If AI is necessary for a nation to stay economically competitive they will figure out a way to make it happen. It's not a huge leap from labor/immigration data to estimating there will be x-number of super-intelligent AI needed to support blah-blah industry.
Your characters have a past that informs who they are, and what kind of choices they've made. Individuals will be either products of society's norms or at odds with it.
Specialized Intelligence, deficits elsewhere
It's a good idea to build-in limitations to superpowers, and also costs – especially since these are some of your POV characters. Design their blindspots and special needs. Do they have a body? Do they need to hire someone to carry them around?
All characters need a relationship with the in-world economy. I suggested above that an AI might 'retire' from their purpose-career after being replaced by newer models (relatable). They might object to their purpose job, or miss it, or maybe they were only mediocre at it (relatable). Anchor them in the 'real' world by showing us how they are supporting their energy bills and upkeep (relatable).
Show, don't tell
You'll need to establish how their intelligence works before you can use it to solve story conflicts. That's why giving them a job is a good idea, we can see their 'normal' everyday use of the superpower, and get an idea of their moral compass – who benefits from their intelligence?
These moral parameters are probably more important to the reader than their lore, or creating a gee-whiz moment with technobabble. A POV character is a protagonist. They see the world in a certain way, and whether they save the cat or kick the dog is how stories work (AI or not).
Show how they apply their intelligence, don't just tell us how smart they are with extra words or by thinking smug thoughts (Sherlock).
The endgame of being the smartest person in the room, is having considered all the possibilities and covered all the bases. TV Tropes calls this the Xanatos Gambit, where a villain has out-maneuvered every other player by reaping some benefit no matter the outcome. That works best when it turns out the villain (Xanatos) was instigating the conflict all along, and it's a final reveal.
Scaling that idea down, the super-intelligent AI will recognize the chaos of putting a real plan into motion, and they will have contingency plan b, and then contingency plan c. Planning for possible outcomes that no one else sees is going to become expensive, and inefficient. It's also a story bore because there are no stakes (much like broken time travel, the hero can always have retro-fixed any problem in the past). This is another superpower that needs limits unless they are the villain.
A 'good' mastermind might try the reverse, to engineer a path where everyone comes out ahead, everyone benefits, no matter the outcome – or they may create the illusion of a big win for their employer meanwhile re-routing resources and protecting the victims in some way.