How to identify the root note of a chord

Answer 
The root note is always the note that is the basis for the chord, regardless of its inversion. In root position the lowest note is the root (hence the name), but other notes are the lowest in other inversions of the chord.
For example, take a C Major chord. In every position, the root note is C. Whether it is voiced as C-E-G (root position), E-G-C (first inversion), or G-C-E (second inversion) is irrelevant. (The voicing is always given from the lowest note to the highest.)
There are multiple ways to interpret chords as well. For example, C-E-A could be considered a C Major 6th in root position or an A minor chord in first inversion. How you interpret it depends upon its function in the song as well as the song's key. In the key of A minor you would definitely consider it to be an A minor chord.
C-E-A could also be considered a rootless F Major 7th chord (normally F-A-C-E) in second inversion. You might interpret it this way in certain F Major pieces, for example. In this case you might say it has no root, or that the root is an "implied" F.

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