To simplify this expression, we need to first find a common denominator for the fractions and then perform the operations.

1 1/2 = 3/2

Now we have:

((3/2)^3 - 3/4) / 7/8

(27/8 - 3/4) / 7/8

To subtract the fractions, we need to find a common denominator:

27/8 - 3/4 = 27/8 - 6/8 = 21/8

Now we have:

21/8 / 7/8

To divide by a fraction, we multiply by the reciprocal:

21/8 * 8/7 = 21/7 = 3

Therefore, the final answer is 3.