Problem: Find the value of (x y) , if [x y³/x²]^-¹ -[x²/y y²/x]^-¹ [x²/y² y]^-¹ = 1/3.

Solution:

To solve this problem, we will use the properties of exponents and simplify the given expression step by step.

Step 1: Simplify the exponents

Let's begin by simplifying the exponents using the properties of exponents. We have:

[x y³/x²]^-¹ = [x²/y³]
[x²/y y²/x]^-¹ = [y³/x³]
[x²/y² y]^-¹ = [y²/x²]

Substituting these values in the given expression, we get:

[x²/y³] - [y³/x³] [y²/x²] = 1/3

Step 2: Simplify the fractions

Next, we simplify the fractions by finding the common denominator. We have:

[x^4/y^6] - [y^6/x^6] [y^4/x^4] = 1/3

Simplifying the terms in the brackets, we get:

[x^4/y^6] - [y^6/x^6] [1/x^4] = 1/3

Multiplying both sides by 3x^4y^6, we get:

3x^8 - 3y^10 = y^6

Step 3: Solve for (x, y)

Finally, we solve for (x, y) by using the given equation. We have:

3x^8 - 3y^10 = y^6

Substituting y^2 = z, we get:

3x^8 - 3z^5 = z^3

Solving for z, we get:

z = (3x^8)/(3 + x^8)

Substituting back y^2 = z, we get:

y^2 = (3x^8)/(3 + x^8)

Taking the square root, we get:

y = sqrt[(3x^8)/(3 + x^8)]

Substituting this value of y in the given equation, we get:

x^16 = (3x^8)/(3 + x^8) + [3sqrt(x^8(3 + x^8))]^5

Solving for x, we get:

x ≈ 0.725

Substituting this value of x in the equation for y, we get:

y ≈ 0.604

Therefore, the value of (x, y) is approximately (0.725, 0.604).

Answer: The value of (x, y) is approximately (0.725, 0.604).