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You could see in every dictionary that cheer and root are synonyms.

cheer (n) = shout of approval, encouragement, or congratulation.
cheer (v) = to salute or acclaim with cheers, applaud

There are the following synonyms: applaud, cheer, root

These verbs mean to express approval or encouragement audibly.

We cheered when the home team scored.
We are rooting for the underdog in the tennis championship.

If I were you, I would said, wit context and without context the both words are complete interchangeable.

In addition you may have yet some synonyms: acclaim, applaud, brighten, clap, console, elate, encourage, hail, hearten, incite, uplift, warm.

V.