And if the number is not a perfect cube, you can try different decimals and adjust until you get an accurate enough approximation or us a special algorithm.

If you just need to simplify cube-root expressions yo just need to find the largest cube y³ that is a factor of n and write n = y³x. Then Cuberoot[n] = y*Cuberoot[x]. Example: Cuberoot[192] = Cuberoot[64*3] = Cuberoot[4³*3] = 4*Cuberoot[3].

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Example of the nth root algorithm applied to approximating cube-root of 75834.632

See if you can follow this:
  4   2.    3   2 ...

75834.632000000...

64 = 4^3

11834

10088 = 42³-40³

  1746632

  1598967 = 423³-420³

    147665000

    107408168 = 4232³-4230³

      40256832 etc.

In order to find the cubed root, all you have to do is find what number times itself three times equals the original number. For example, if you wanted the cubed root of 8. You would go through to find that you multiply the number 2 three times (2x2x2) and you would end up back at your original number (8). So the cubed root of 8 would be 2.