To solve this equation, we can rewrite 32^n as (2^5)^n and 12^-n as (2^2*3)^-n.

So, the equation becomes (2^5)^n * (2^2*3)^-n = 1024.

Using the properties of exponents, we can simplify this equation to 2^(5n) * 2^(-2n*3) = 1024.

Now, we can combine the exponents by adding them together: 5n - 6n = 1024.

Simplifying further, we get -n = 1024.

Finally, we can solve for n by multiplying both sides of the equation by -1: n = -1024.

Therefore, the value of n that satisfies the equation is -1024.