How do I determine what amount of a college scholarship paid for room and board vs. tuition? It is not demarcated on the college invoice.

Your 1098-T box 1 minus box 5 is the amount not spent on tuition and would be taxable.

If your scholarship is not restricted (it must be used for tuition), you get to decide how much of it was used for room & board.  If it was not itemized on the school invoice, the scholarship is, most likely, not restricted. 

You do that to "free up" QEE-Qualified educational expenses (tuition, fees and book expenses) to be claimed for the tuition credit and/or a 529 plan distribution.

Scholarships used for QEE are tax free. Scholarships used for room & board are taxable income. However, it is frequently better to pay a little tax (usually little or none) on the scholarship in order to claim the tuition credit.

For more specific advise, tell us what you're trying to do and whether you are a dependent.