Find the distance between the points with polar coordinates (4, π/3) and (8, 2π/3).

A suitable vector calculator can perform this arithmetic directly, as can many graphing or scientific calculators that are capable of handling complex numbers. Mine says
  (8, 2π/3) - (4, π/3) = (4√3, 5π/6)
The distance is 4√3 ≈ 6.9282.

If you cannot use your calculator for this purpose, you can use the Law of Cosines. You have two sides of the triangle (4 and 8) and the angle between them (2π/3 - π/3), so you have all the information needed.
  c² = a² + b² -2ab·cos(C)
  c² = 4² + 8² - 2·4·8·cos(π/3)
  c² = 16 + 64 - 32 = 48
  c = √48 = 4√3

Or, you can simply recognize that the two vectors have the ratio 1:2 and the angle between them is 60°. That is, they are one leg and the hypotenuse of a 30°-60°-90° triangle, so the other leg is 4√3.