We investigate the problem of approximating function f in the power-type weighted variable exponent Sobolev space W r,p(.) α(.) (0, 1), (r = 1, 2, ...), by the Hardy averaging operator A f (x) = 1 x x 0 f (t)dt. If the function f lies in the power-type weighted variable exponent Sobolev space W r,p(.) α(.) (0, 1), it is shown that A f − f p(.),α(.)−rp(.) ≤ C f (r) p(.),α(.) , where C is a positive constant. Moreover, we consider the problem of boundedness of Hardy averaging operator A in power-type weighted variable exponent grand Lebesgue spaces L p(.),θ α(.) (0, 1). The sufficient criterion established on the power-type weight function α(.) and exponent p(.) for the Hardy averaging operator to be bounded in these spaces.