Either you use a scientific calculator, or you try which number, when cubed, gives 64. This one is easy, since it is a perfect cube. Try, for example, 2 cubed (2 x 2 x 2); this is 8, which is too little, so you try a larger number.
unless it is a perfect cube one cannot do cube roots by hand without knowing calculus. Perfect cubes are easy to recognize with a bit of practice. one knows that 3*3*3 is 27 therefore the cube root...