I was given a table and instructed to fill in the blanks. I cannot figure out how to find the missing values. I have thought, searched, cried, and searched some more.
2006 2007 2008
Ret earnings (beg) $(1746.5) $(1830.50) $(2653.00)
revenues 4840.5 5327.00 ?
expenses ? 5628.00 5425.0
dividends -0- ? 17.5

Please help! I have read the book, watched the lecture, and searched until my fingers hurt.

For this purpose, Retained Earnings is similar to a bank account. While not actually the same, Retained Earnings and bank accounts do have something in common:

The beginning balance (at any given period, such as the first of the month),
Plus the deposits,
Minus the withdrawals,
Equals the ending balance (at the end of that month, say).

If you know any three of those four, you can easily determine the fourth. For example, if I told you I had $100 in an account at the beginning of a year, I deposited $200 during that year, and had $50 in that account at the end of that same year, you can immediately see that I must have withdrawn $250 during the year.

Now for Retained Earnings, revenues are the deposits, while expenses and dividends are the withdrawals*.

*(You might hear that net income is the increase to retained earnings, while net loss and dividends are the decreases, but that's all the same thing, as net income / net loss is just Revenues minus Expenses.)

Check your given facts carefully, though. If those three retained earnings figures are indeed beginning numbers (for 2006, 2007, 2008), then without knowing the end-of-2008 (equivalently, beginning-of-2009) retained earnings, you cannot solve for 2008 Revenues.