  We treat two different equations involving powers of singular moduli. On the
one hand, we show that, with two possible (explicitly specified) exceptions,
two distinct singular moduli j(\tau),j(\tau') such that the numbers 1,
j(\tau)^m and j(\tau')^n are linearly dependent over $\mathbb{Q}$ for some
positive integers m,n, must be of degree at most 2. On the other hand, we show
that, with obvious exceptions, the product of any two powers of singular moduli
cannot be a non-zero rational number.
