We provide a method to evaluate certain Dedekind eta quotients by finding a modular relation of order p which involves the j-function. The relation can be explicitly provided whenever p-1 divides 24. Ramanujan's eta-quotients involve only when p = 3, but we provide the family of functions for prime order p = 2, 3, 5, 7, 13.
As a happy consequence of these evaluations, new approximations to e^(pi*sqrt(d)) can be found, one of which is,
e^(pi*sqrt(163)) ~ 5^3(5x^6-640320x^5-10x^3+1)^3 - 6.0000000000...
using the appropriate real root of the sextic equated to zero. The difference is not exactly 6, of course, but very close.
Certain identities obeyed by these functions analogous to Weber's octic identity are also given: one, for order 3, is a sextic analogue and another, for order 5, is a multi-grade identity.
