\frac{a x-a y+b x-b y}{x^2-y^2} \times \frac{x^2 y+x y^2}{a b^2+a^2 b}
Answer (2)
The expression simplifies to ab x y through a series of factorizations and cancellations. First, each part of the original expression is factored, allowing for the cancellation of common factors. This results in a much simpler form of the expression, which is useful in various mathematical applications.
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Factor the numerator of the first fraction by grouping: a x − a y + b x − b y = ( a + b ) ( x − y ) .
Factor the denominator of the first fraction as a difference of squares: x 2 − y 2 = ( x − y ) ( x + y ) .
Factor the numerator of the second fraction: x 2 y + x y 2 = x y ( x + y ) .
Factor the denominator of the second fraction: a b 2 + a 2 b = ab ( a + b ) . Simplify the expression to get the final answer: ab x y .
Explanation
Understanding the Expression We are asked to simplify the expression x 2 − y 2 a x − a y + b x − b y × a b 2 + a 2 b x 2 y + x y 2 . This involves factoring and cancelling common terms.
Factoring the First Numerator First, let's factor the numerator of the first fraction by grouping: a x − a y + b x − b y = a ( x − y ) + b ( x − y ) = ( a + b ) ( x − y ) .
Factoring the First Denominator Next, factor the denominator of the first fraction using the difference of squares: x 2 − y 2 = ( x − y ) ( x + y ) .
Factoring the Second Numerator Now, factor the numerator of the second fraction: x 2 y + x y 2 = x y ( x + y ) .
Factoring the Second Denominator Factor the denominator of the second fraction: a b 2 + a 2 b = ab ( b + a ) = ab ( a + b ) .
Rewriting the Expression Rewrite the expression with the factored terms: ( x − y ) ( x + y ) ( a + b ) ( x − y ) × ab ( a + b ) x y ( x + y ) .
Cancelling Common Factors Now, cancel out the common factors in the numerator and denominator: ( x − y ) ( x + y ) ( a + b ) ( x − y ) × ab ( a + b ) x y ( x + y ) = ( x + y ) ( a + b ) × ab ( a + b ) x y ( x + y ) = ab x y .
Final Answer Therefore, the simplified expression is ab x y .
Examples
Simplifying algebraic expressions is useful in various fields, such as physics and engineering, where complex formulas can be made easier to understand and work with by simplifying them. For example, in circuit analysis, simplifying expressions involving resistances and currents can help in designing efficient circuits. In physics, simplifying equations of motion can help in predicting the trajectory of objects. This skill is also crucial in computer science for optimizing code and algorithms.
