Three consonants for one human sound: laughter. The plain verb laughs, the causative makes others laugh, and the noun is laughter itself. This root also hides inside a name every Hebrew speaker knows: יִצְחָק (Isaac) means "he will laugh", Sarah laughed when she was told she would bear a son. So the root carries both the giggle and a piece of Genesis. 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.
The contrast is clean: pa'al לִצְחֹק is you laughing, hif'il לְהַצְחִיק is causing the laugh in someone else, exactly what the causative voice does. The pi'el לְצַחֵק ("to mock / sport with") is mostly biblical today. The passive partners pu'al and huf'al and the reflexive hitpa'el are not lexicalized here.
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.