​If xy(x+y)=1, then the value of ​$$1/(x^3y^3 )-x^3-y^3$$ is​

Correct option is A

We are tasked with solving the expression $$\frac{1}{x^3y^3} - x^3 - y^3$$​ given xy(x + y) = 1 .

Step-by-Step Solution:

1. Given Equation:

xy(x + y) = 1

2. Key Expression:

We need to evaluate:

​$$\frac{1}{x^3y^3} - x^3 - y^3$$​

3. Substitute:

From the given equation, multiply both sides by $$(xy)^2$$​ :

​$$(xy)^3 = 1$$​

Hence:

xy = 1

4. Simplify the Expression:

Substituting xy = 1 into $$\frac{1}{x^3y^3}$$​ :

​$$\frac{1}{x^3y^3} = \frac{1}{(xy)^3} = \frac{1}{1} = 1$$​

Now, the expression becomes:

​$$\frac{1}{x^3y^3} - x^3 - y^3 = 1 - x^3 - y^3$$​

5. Relation Between $$x^3 + y^3$$​ :

Using the identity for $$x^3 + y^3$$​ :

​$$x^3 + y^3 = (x + y)((x + y)^2 - 3xy)$$​

Since xy = 1 , this simplifies to:

​$$x^3 + y^3 = (x + y)((x + y)^2 - 3)$$​

6. Substitute Back:

From the given equation xy(x + y) = 1 , we know:

x + y = 1

Substitute x + y = 1 into $$x^3 + y^3$$​ :

​$$x^3 + y^3 = 1((1)^2 - 3) = 1(1 - 3) = -2$$​

7. Final Simplification:

Substituting $$x^3 + y^3 = -2$$​ into the expression:

​$$\frac{1}{x^3y^3} - x^3 - y^3 = 1 - (-2) = 1 + 2 = 3$$​

Final Answer:

​$$\boxed{3}$$​