This is the root of the number one, אֶחָד, the word every Hebrew speaker says in the Shema. Stretch those three consonants into a verb and you stop counting and start acting: making things one, merging them, joining together. The nouns then name the result of that merging: unity, a federation, a union. Knowing אֶחָד already hands you half the family. Every word here is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
No pa'al here: "one" is a state, not a basic action, so the family starts at pi'el (make one). Pi'el unites, the causative hif'il standardizes (forces things into one mould), and the reflexive hitpa'el is the group uniting itself.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.