If you're just trying to graph the equation, use something like Desmos or Wolfram|Alpha. These sort of general-purpose tools are much more powerful than graphing calculators, and can graph almost anything you can think of, including implicit equations.
If you want to graph on the TI-83+ specifically, you'll need to do some easy math:
x² + (y-x^(2/3))² = 1
(y - x^(2/3))² = 1 - x²
y - x^(2/3) = ±√(1 - x²)
y = x^(2/3) ± √(1 - x²)
Now you have Y in terms of X. The TI-83+ series doesn't have a ±
sign, though, so you'll need to graph the two equations as a list. Type this in in Y=
mode:
X^(2/3)+{1,-1}√(1-X²)