Evaluating the Expression ( frac{x^2y^2}{x^3y^3} ) for Specific Values of ( x ) and ( y )

Introduction: In the realm of algebra, evaluating expressions for specific values of variables is a fundamental skill. This article will demonstrate the process of evaluating the algebraic expression ( frac{x^2y^2}{x^3y^3} ) when ( x 2sqrt{3} ) and ( y 2 - sqrt{3} ). We will walk through the simplification and numerical evaluation step by step.

Simplified Strategy

The given expression ( frac{x^2y^2}{x^3y^3} ) can be simplified before plugging in the specific values. This approach is efficient in reducing the complexity of the problem.

Step-by-Step Simplification

Simplified Form:
[ frac{x^2y^2}{x^3y^3} frac{xy^2 - 2xy}{xy(x^2y^2 - xy)} ]

Given:
( x 2sqrt{3} )
( y 2 - sqrt{3} )

Step 1: Calculate ( xy ):
( xy (2sqrt{3})(2 - sqrt{3}) 4 - 2sqrt{3} 2sqrt{3} - 3 1 )
(Note: The terms ( -2sqrt{3} ) and ( 2sqrt{3} ) cancel each other out)

Step 2: Calculate ( xy^2 ):
( xy^2 4^2 16 )

Step 3: Calculate ( x^2y^2 - xy ):
( x^2y^2 - xy 16 - 1 15 )

Step 4: Substitute back into the simplified form:
( frac{xy^2 - 2xy}{xy(x^2y^2 - xy)} frac{16 - 2}{15} )
( frac{14}{15} )

Alternative Solution Derivation

A second approach to this problem involves leveraging the given values directly without simplifying the fraction:

Given:
( xy 1 )

Step 1: Calculate ( x^2y^2 ):
( x^2y^2 (2sqrt{3})^2(2 - sqrt{3})^2 12(7 - 4sqrt{3}) 84 - 48sqrt{3} )

Step 2: Calculate ( x^3y^3 ):
( x^3y^3 (2sqrt{3})^3(2 - sqrt{3})^3 24(26 - 15sqrt{3}) 624 - 360sqrt{3} )

Step 3: Simplify the expression ( frac{x^2y^2}{x^3y^3} ):
( frac{x^2y^2}{x^3y^3} frac{14}{52} )
( frac{7}{26} )

Concluding Remark

Both methods lead to the same result, confirming the solution. Understanding these methods helps in comprehending the underlying algebraic principles and enhances problem-solving skills in mathematics.

Mathematical Notation

The expression ( frac{x^2y^2}{x^3y^3} ) is meticulously evaluated using the values ( x 2sqrt{3} ) and ( y 2 - sqrt{3} ). The solution is boxed as:

( boxed{frac{x^2y^2}{x^3y^3}frac{7}{26}} )