The purpose of this paper is to show that 3ⁿ annihilates the 3-primary component of the homotopy groups of the 2n + 1-dimensional sphere. In the terminology of (2) and (3), S²ⁿ⁺¹ has exponent 3ⁿ at 3.In fact, a stronger result is proved. Localize at 3 and let Ω²ⁿS^{2n + 1}〈2n + 1〉 denote the 2n-fold loop space of the 2n + 1-connected cover of S²ⁿ⁺¹. Then Ω²ⁿS^{2n + 1}〈2n + 1〉 has a null homotopic 3ⁿ-th power map.