the other one came out wrong.

8x^2y^-2 (4xy^2)^-1
________ . ___________
x^-2y x^2y

its a fraction and you have to multiply

I assume you mean
[(8x^2 y^-2)/(x^-2 y)] x [(4xy^2)^-1/(x^2 y)]
which can be rewritten
= 8 x^4 y^-3 x [1/(4x^3y^3)]
= 2 x y^-6

To simplify the expression, you can follow these steps:

Step 1: Simplify each part of the fraction separately.
In the numerator:
- Simplify 8x^2y^-2 by using the property of exponents. Since y^-2 means 1/y^2, the numerator becomes 8x^2/(1/y^2).
- Simplify (4xy^2)^-1 by using the property of exponents. Since (4xy^2)^-1 means 1/(4xy^2), the numerator becomes 1/(4xy^2).

In the denominator:
- Simplify x^-2y by using the property of exponents. Since x^-2 means 1/x^2, the denominator becomes (1/x^2)y.
- Simplify x^2y by using the property of exponents. This remains the same.

Step 2: Multiply the resulting numerator and denominator fractions.
- Multiply (8x^2/(1/y^2)) by (1/(4xy^2)) to get (8x^2/(1/y^2)) * (1/(4xy^2)) = (8x^2)/(1/y^2 * 4xy^2).
- Multiply ((1/x^2)y) by (x^2y) to get ((1/x^2)y) * (x^2y) = ((1/x^2)y * x^2y).

Step 3: Simplify the resulting fractions.
- Simplify (1/y^2 * 4xy^2) by canceling out common factors. In this case, y^2 cancels out.
- Simplify ((1/x^2)y * x^2y) by canceling out common factors. In this case, x^2y cancels out.

Step 4: Multiply the remaining terms.
- Multiply (8x^2)/(1/4x) to get (8x^2) * (4x) = 32x^3.

Therefore, the simplified expression is 32x^3.