Breakdown of Ég læt hárgreiðslumanninn klippa hárið.
ég
I
láta
to let
hárið
the hair
hárgreiðslumaðurinn
the hairdresser
Questions & Answers about Ég læt hárgreiðslumanninn klippa hárið.
Why is hárgreiðslumanninn in the accusative case here?
This is the core of the causative construction you are learning. The verb láta (here as læt) takes an accusative object for the person being made or allowed to do the action. So, the hairdresser (hárgreiðslumaðurinn) changes to the accusative form hárgreiðslumanninn.
The translation says "I let the hairdresser cut...", but wouldn't "I have the hairdresser cut..." make more sense?
Yes, it absolutely would! In Icelandic, láta is used for both letting (allowing) and making or having someone do something (causing). Context usually tells you which one it is. Since you normally arrange a haircut, "I am having the hairdresser cut my hair" is the most natural English understanding here.
Why does it say hárið (the hair) instead of my hair?
In Icelandic, when talking about body parts or things closely associated with yourself, you usually just use the definite article rather than a possessive pronoun like mitt. Because it is obvious whose hair is being cut, hárið naturally translates to "my hair" in this context.
How do you break down the long word hárgreiðslumanninn?
It is a compound noun: hár (hair) plus greiðsla (combing or styling) plus manninn (the man or person, in the accusative case). So a hárgreiðslumaður is literally a hair-styling-person.