(x^−2/x^1)p · (y^q/x)2=xy

Simplify the following exponential equation to determine the values of p and q.

Your notation is unclear. It appears that you want to say

(x^−2/x^1)^p · (y^q/x)^2=xy
(x^-3)^p (y^q/x)^2 = xy
x^-(3p+2) y^(2q) = xy

equating powers, we have

3p+2 = -1
p = -1

2q = 1
q = 1/2

To simplify the equation and determine the values of p and q, we can break it down step by step as follows:

1. Start with the given equation:
(x^(-2)/x^1)p · (y^q/x)^2 = xy

2. Simplify the terms within the parentheses using the exponent rules:
(x^(-2)/x^1)p = (1/x^2)p = 1^p/x^(2p)
(y^q/x)^2 = (y^2q)/x^2

3. Substitute the simplified terms back into the equation:
(1^p/x^(2p)) · (y^2q/x^2) = xy

4. Combine the terms with the same base (x) in the numerator:
(1^p · y^2q)/(x^(2p) · x^2) = xy

5. Simplify further:
1^p = 1, so we can remove it from the equation.

(y^2q)/(x^(2p) · x^2) = xy

6. We can simplify the equation further by combining the x terms in the denominator:
(y^2q)/(x^(2p + 2)) = xy

7. Now, we can compare the exponents on both sides of the equation:
2q = 1 (from y^2q = x^1)
2p + 2 = 1 (from x^(2p + 2) = x^1)

8. Solve the equations:
From 2q = 1, we see that q = 1/2.

From 2p + 2 = 1, subtracting 2 from both sides gives:
2p = -1, and dividing by 2 gives:
p = -1/2.

Therefore, the values of p and q that satisfy the given equation are p = -1/2 and q = 1/2.