It is shown in this paper that 1, \({{2(k-1)} \over {2k-1}}\), \({{2(k-1)} \over {2k-1}}\) form admissible exponents for three \(k\)-th powers. This improves Davenport’s results for \(5\leq k\leq 8\), and H. Davenport and P. Erdős’s result for \(k\geq 9\) [Ann. Math., II. Ser. 40, 533-536 (1939; Zbl 0021.20703)].