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Carpi (1993) and Lepisto (1994) proved independently that smooth words arecube-free for the alphabet {1, 2}, but nothing is known on whether for theother 2-letter alphabets, smooth words are k-power-free for some suitablepositive integer k. In this paper, we first establish the derivative formula ofthe concatenation of two smooth words and power derivative formula of smoothwords over arbitrary 2-letter alphabets. Then by making use of power derivativeformula, for arbitrary 2-letter alphabet {a, b} with a, b being positiveintegers and a<b, we prove that smooth words are (b+1)-power-free except fora=1 and b=3; and smooth words are quintic-free and there are infinitely manysmooth biquadrates for the alphabet {1, 3}. Moreover, we give the number ofsmooth words of form w^n with a and b having the same parity.

The powers of smooth words over arbitrary 2-letter alphabets