Hi James,
If I am correctly interpreting your question, it looks like you are multiplying two power expressions to get a certain result and you need to determine the values of the constants, a & b, in order to achieve this result. Before we begin to examine your problem, lets look at a simpler version of the same type of question:

(6*x²*y³)(-2*x*y⁴) = -12*x³*y⁷ We multiply the coefficients together (6*-2 = -12)
We add the exponents on the x (2 + 1 = 3)
We add the exponents on the y (3 + 4 = 7)

Now that we have a process to follow, let's look at your problem:

(ax^-1*y)(-1/2xy^b) = 6y^-6 Let's multiply the left-hand side to see what it would look like.

a*(-1/2)*x^-1+1*y^1+b = 6y^-6 Simplify what you can.

-a/2 * x⁰ *y^1+b = 6y^-6 But x⁰ = 1 so we don't need to write it.

-a/2*y^1+b = 6y^-6 Let's see what the "a" and "b" match up with on the other side.

To determine the value of "a", we need to solve the following equation that equates the coefficients from both sides:

-a/2 = 6 solve this and we get a = -12

To determine the value of "b", we need to solve the following equation that equates the exponent on y from both sides:

1 + b = -6 solve this and we get b = -7

Try substituting these values into your original problem and see if it does yield the result you need.

Hope this was clear enough for you.
Cheers!

Leeanne