There are two schools of thought on when the Exodus occurred, the "early" date of 1440BC, and the "later" date of 1290BC. How was each date arrived at, and what is the evidence for each?
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The method of deducing each date, along with the difficulties associated with each method, are documented at cresourcei.org
The older date (1440 BC) is primarily based on the assumption that the Bible is a reliable historical document, with the passage of time based solely upon dates and time periods given from within Scripture. Example:
The more recent date (1290 BC) is primarily based on external archaeological and external evidence.
These are not the only evidences for each date. It should be noted that there are various assumptions and difficulties with either date. I am not about to list them here, as they are all well documented on the page I linked to above. Suffice it to say that the evidence for each is seen as tenuous, at best by those that hold the opposite belief.
Also, in the sections above, note that I said that each position is based primarily on either internal Biblical evidence or external evidence. That does not mean the position is based only on those two things. Each position is just as much based on an understanding of the difficulties of the opposing view.
Which date we hold to is likely to fall along the lines of how much we trust the Bible for historical accuracy, and how much we trust our interpretation of it, as well as where we draw the line on literalism. (Many of the arguments for a more recent date point out scenarios where a given time span can be interpreted as something other than the direct literal reading. Sounds like another debate about dating certain events that we're all quite familiar with.)