One way to look at this is to take two radicals: square root of a and cube root of b, which differing indices. But we can introduce the LCM of 2 and 3 which is 6 and write √a=(6)√(a^3) and (3)√b=(6)√(b^2) where the number in parentheses is the index.This introduces a common radical–6th root. Now we multiply under the same index: (6)√(a^3b^2). So in the general case we need the LCM of the indices as the common radical and then we use the power indices under the common radical. So it is not necessary for 2 radical expressions to have the same index in order to multiply them.
NUMERICAL EXAMPLE: square root of 121 and cube root of 27. We know the answer is 11*3=33. But let's see if we get the same result by putting a=121 and b=27 in the formula above. The product should be (6)√(177561*729)=(6)√1291467969=33. So the formula works.