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Until you get good with exponents it will help if you think of exponents in factored form.
I can factor -3/2 into -1*3*(1/2).
There are a couple of advantages to looking at the exponent in factored form:
We can see the "operations" that are represented. With this exponent, let's look at each factor:
-1 means a reciprocal will occur
3 means cubing will occur
1/2 means a square root will occur
Since multiplication is Commutative we can use the factors in any order we choose.
So we are going to do a reciprocal, a cube and a square root in any order we choose. Let's choose an order that make things easier.
Reciprocals are not difficult and cubing is not difficult either. However, since 8 is not a perfect square. a square root at the start will not make things easy. So we'll start with either a reciprocal or cubing. Since a reciprocal means a fraction, I'll postpone that for later and cube 8 first: We still have a reciprocal and a square root to do. 512 is not a perfect square so I'll leave that for last. The reciprocal of 512 is 1/512.
The only factor of the exponent left is the square root left so we end up with
The only thing left is to simplify (including rationalizing the denominator). There are different ways to do this. One way, which keep the numbers smaller and easier to work with, follows:
Using the property of radicals, , we can split into a fraction of square roots:
Simplifying the two square roots:
Rationalizing the denominator:
(