I wanted to know what's the exact difference between 住 and 居.
Since these characters mean the same thing, why can't I convince them together into one word meaning "to live"?
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 [v] live; reside; stay; dwell; inhabit  stop; cease
 [v] to live at, dwell, reside, inhabit, occupy, abode  [n] a residence, an house, dwelling
to reside; to live (at some place) | [粵]: 住
Since the meaning of " to live / to reside " has a dedicated verb 住 , and 居 as a verb, has other meanings (for example: occupy) , few people would use 居 for the meaning of 住 , except in instances that literal style of reading is preferred.
You can think of 居 is more literal and 住 is more colloquial when it comes to using a verb for "to live ; to reside"
Some example sentences:
他現在住在美國 (he currently resides in America)
他現居於美國 (he currently resides in America)
他現在在美國居住 (he currently resides in America)
*unlike 住 or 居, which is placed before the object , 居住 is placed after the object.
I don't have enough points to comment yet, but 居住 can be used before the object.
他居住在美国 is both acceptable and correct.
住 means live
居住 also means live but contains the meaning of stay (the period is longer than 住), and can imply "have a life there", such as set up a home. Even it doesn't have to be permanent, the length of time cannot be very short. Here are some examples:
a. You can say my friend is going to live with me/ in my house for 3 days：我朋友要在我家住三天。You cannot say 我朋友要在我家居住三天。This is because 3 days is a short period of time.
b. You can say you lived in a hotel for 3 days: 我在旅馆里住了三天。You cannot use 居住 in this case (lived in a hotel for 3 days) because you only live there temporarily and it doesn't require you to set up a home.
c. You can say 他居住在美国三年了。 As 3 years is a relatively long period and it requires you to set up a home of some kind to live in a city/country for this long. 居住 is more appropriate in this case.
住and居 are interchangeable, and they can also be used together, with slightly different in meaning.
住者人主也**。 人 means human, person; 主 means occupy, owner. Thus 住 essentially means living in some place for a relative short term.
居者尸古也**。 尸 means corpse, dead body; 古 means ancient, old or long period of time here. Thus 居 essentially means living in some place too, but for a long term because ever after you are dead, ie, becoming a 尸 corpse, your soul will still live in that place for a long period of time.