Powers of Euler's product and related identities

Ramanujan's partition congruences can be proved by first showing that the c oefficients in the expansions of (q; q)(infinity)(r) satisfy certain divisi bility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r, the coefficients in the expansions of (q; q)( infinity)(r) satisfy arithmetic relations, and these arithmetic relations i mply the divisibility properties referred to above. We also obtain arithmet ic relations for the coefficients in the expansions of (q; q)(infinity)(r) (q(t); q(t))(infinity)(s), for t = 2, 3, 4 and various values of r and s. O ur proofs are explicit and elementary, and make use of the Macdonald identi ties of ranks 1 and 2 (which include the Jacobi triple product, quintuple p roduct and Winquist's identities). The paper concludes with a list of conje ctures.