$\overline{x^2-4}$
Simplify:
(a) $\frac{x^2}{y(x-y)}+\frac{y^2}{x(y-x)}$
Answer (1)
Rewrite the expression with a common denominator.
Combine the fractions into a single expression.
Factor the numerator using the difference of cubes formula.
Cancel the common factor ( x − y ) and simplify the expression to y x + 1 + x y .
Explanation
Understanding the Problem We are asked to simplify the expression y ( x − y ) x 2 + x ( y − x ) y 2 .
Rewriting the Expression First, we rewrite the second term to have a common factor in the denominator. Notice that ( y − x ) = − ( x − y ) . So we can rewrite the expression as: 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 − x ( x − y ) y 2 ⋅ y y = x y ( x − y ) x 3 − x y ( x − y ) y 3
Combining Fractions Combine the two fractions: x y ( x − y ) x 3 − y 3
Factoring the Numerator Now, we factor the numerator using the difference of cubes formula: x 3 − y 3 = ( x − y ) ( x 2 + x y + y 2 ) . So the expression becomes: x y ( x − y ) ( x − y ) ( x 2 + x y + y 2 )
Simplifying the Expression We simplify the expression by canceling the common factor ( x − y ) from the numerator and the denominator: x y ( x − y ) ( x − y ) ( x 2 + x y + y 2 ) = x y x 2 + x y + y 2
Rewriting as a Sum of Fractions Rewrite the expression as a sum of fractions: x y x 2 + x y x y + x y y 2
Simplifying Each Fraction Finally, simplify each fraction: y x + 1 + x y Thus, the simplified expression is y x + 1 + x y .
Examples
This simplification skill is useful in various fields, such as engineering, physics, and computer science, where complex equations need to be simplified for easier analysis and computation. For example, when analyzing electrical circuits, simplifying complex impedance expressions can make calculations easier. Similarly, in fluid dynamics, simplifying equations involving flow rates and pressures can help in designing efficient systems. In computer graphics, simplifying equations for transformations and projections can optimize rendering performance.
