Fact 1: Graph x raised to the 1/3 and get a graph that runs in the 1st and 3rd quadrant. 
The domain of f(x) includes negative numbers because x raised to any odd power is negative and the cube root of any negative number exists.
You are the master, the calculator is the slave. It is up to you to be smart so that the calculator will do what you intend it to do... nothing more and nothing less!
Does anyone have a suggestion why the number of repeating digits should change the appearance of the graph?
Fact 1: Graph x raised to the 1/3 and get a graph that runs in the 1st and 3rd quadrant.
The domain of f(x) includes negative numbers because x raised to any odd power is negative and the cube root of any negative number exists.
Fact 2: Graph x raised to the .333 the graph is in the 1st quadrant only (as the book suggests it should).
The domain of g(x) does not include negative numbers because x raised to any odd power is negative and no even roots of negative numbers exist. Even roots of x raised to 0.3, 0.33, 0.333, 0.3333, 0.33333, etc. do not exist for negative x for the same reason.
Fact 3: Graph x raised to the .333333333333333333 we get the graph back in the 1st and 3rd quadrant.
At some point (intentional pun) floating point calculators run out of decimal places to the right. At that point, the calculator has been programmed by people to round off repeating decimals to the nearest fraction, one third in this case. See response to Fact 1 to see why roots of negative x "reappear".