The ratio is the same as the fraction √6/(64x9y9z15).
64=26, so 64x9y9z15=26x9y9z15=(22x3y3z5)3=(4x3y3z5)3.
One simplification is therefore:
√6:(4x3y3z5)3, or √6:(4(xy)3z5)3.
[We could take this further because √6=6/√6 making the ratio:
6:(4(xy)3z5)3√6, which doesn't look simpler. Also 43√6=43¼√3, but that also looks more complicated.]