Simplify:
(a) $\frac{x^2}{y(x-y)}+\frac{y^2}{x(y-x)}$
Answer (2)
Rewrite the expression with a common denominator: y ( x − y ) x 2 − x ( x − y ) y 2 = x y ( x − y ) x 3 − y 3 .
Factor the numerator using the difference of cubes: x y ( x − y ) ( x − y ) ( x 2 + x y + y 2 ) .
Cancel the common factor ( x − y ) : x y x 2 + x y + y 2 .
Rewrite as a sum of fractions and simplify: y x + 1 + x y . The simplified expression is 1 + y x + x y .
Explanation
Understanding the Problem We are asked to simplify the expression y ( x − y ) x 2 + x ( y − x ) y 2 . This involves algebraic manipulation to combine the two fractions into a single, simpler expression.
Rewriting the Expression First, notice that ( y − x ) = − ( x − y ) . We can rewrite the second term to have a common factor in the denominator with the first term: y ( x − y ) x 2 + x ( y − x ) y 2 = y ( x − y ) x 2 − x ( x − y ) y 2
Finding a Common Denominator Now, we find a common denominator for the two fractions, which is x y ( x − y ) . We rewrite each fraction with this common denominator: y ( x − y ) x 2 − x ( x − y ) y 2 = x y ( x − y ) x 2 ⋅ x − x y ( x − y ) y 2 ⋅ y = x y ( x − y ) x 3 − x y ( x − y ) y 3 Combining the fractions, we get: x y ( x − y ) x 3 − y 3
Factoring the Numerator We can factor the numerator using the difference of cubes formula, which states that x 3 − y 3 = ( x − y ) ( x 2 + x y + y 2 ) . Applying this to our expression, we have: x y ( x − y ) ( x − y ) ( x 2 + x y + y 2 )
Canceling Common Factors Now, we can cancel the common factor of ( x − y ) from the numerator and the denominator, assuming x = y : x y ( x − y ) ( x − y ) ( x 2 + x y + y 2 ) = x y x 2 + x y + y 2
Simplifying the Expression Finally, we can rewrite the expression as a sum of fractions: x y x 2 + x y + y 2 = x y x 2 + x y x y + x y y 2 = y x + 1 + x y So the simplified expression is 1 + y x + x y .
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics with applications in various fields. For instance, in physics, you might encounter complex equations describing the motion of objects or the behavior of electromagnetic fields. Simplifying these equations can make them easier to analyze and solve, leading to a better understanding of the underlying physical phenomena. Similarly, in engineering, simplifying expressions can help optimize designs and improve the efficiency of systems. For example, consider an electrical circuit with multiple components. Simplifying the equation that describes the circuit's behavior can help engineers determine the optimal values for the components to achieve desired performance characteristics. In computer science, simplifying expressions is crucial for optimizing algorithms and reducing computational complexity. By simplifying code, programmers can improve the speed and efficiency of their programs, making them more responsive and user-friendly. These examples illustrate the broad applicability of simplification skills in diverse scientific and technological domains.
To simplify the expression y ( x − y ) x 2 + x ( y − x ) y 2 , rewrite it using a common denominator, factor the numerator, and cancel common terms. The resulting expression is 1 + y x + x y .
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