Let X be a finite CW-complex. We show that the image of the homotopy groups of X under suspension have an exponent at every prime. As a corollary we recover Long's result that finite H-spaces have exponents at all primes. We show that the stable homotopy groups of X have an exponent at p if and only if X is rationally equivalent to a point. This allows us to construct many examples of spaces with infinite dimensional rational homotopy groups and without an exponent at any prime. These examples further support Moore's conjecture.