\frac{\left(x^2-y^2\right)}{(8 x-2 y)} \frac{(16 x-4 y)}{(x+y)^2}
Answer (1)
Factor x 2 − y 2 to ( x − y ) ( x + y ) .
Factor 8 x − 2 y to 2 ( 4 x − y ) and 16 x − 4 y to 4 ( 4 x − y ) .
Rewrite the expression and cancel out the common factors ( 4 x − y ) and ( x + y ) .
Simplify the expression to get the final answer: x + y 2 ( x − y ) .
Explanation
Understanding the Problem We are asked to simplify the expression ( 8 x − 2 y ) ( x 2 − y 2 ) ( x + y ) 2 ( 16 x − 4 y ) . To do this, we will factor the terms and cancel out common factors.
Factoring the Numerator First, we factor the numerator x 2 − y 2 using the difference of squares formula: x 2 − y 2 = ( x − y ) ( x + y ) .
Factoring the First Denominator Next, we factor out a 2 from the term ( 8 x − 2 y ) : 8 x − 2 y = 2 ( 4 x − y ) .
Factoring the Second Numerator Then, we factor out a 4 from the term ( 16 x − 4 y ) : 16 x − 4 y = 4 ( 4 x − y ) .
Expanding the Second Denominator We expand the denominator ( x + y ) 2 as: ( x + y ) 2 = ( x + y ) ( x + y ) .
Rewriting the Expression Now, we rewrite the expression with the factored terms: 2 ( 4 x − y ) ( x − y ) ( x + y ) ⋅ ( x + y ) ( x + y ) 4 ( 4 x − y ) .
Canceling Common Factors We cancel the common factor ( 4 x − y ) from the numerator and denominator: 2 ( x − y ) ( x + y ) ⋅ ( x + y ) ( x + y ) 4 .
Further Simplification We cancel the common factor ( x + y ) from the numerator and denominator: 2 ( x − y ) ⋅ ( x + y ) 4 .
Final Simplification Finally, we simplify the expression: 2 ( x + y ) 4 ( x − y ) = ( x + y ) 2 ( x − y ) .
Final Answer Thus, the simplified expression is x + y 2 ( x − y ) .
Examples
Simplifying rational expressions is a fundamental skill in algebra that can be applied in various real-world scenarios. For instance, in physics, when dealing with projectile motion, simplifying complex equations involving variables like initial velocity, launch angle, and gravity is crucial for predicting the range and trajectory of a projectile. By simplifying these expressions, physicists can more easily analyze and understand the relationships between different variables, leading to more accurate predictions and a deeper understanding of the physical phenomena involved. Similarly, in engineering, simplifying rational expressions is essential for designing efficient structures and systems, such as bridges and electrical circuits.
