[tex]\frac{x+1}{x^2 y^2} \div \frac{x^2-1}{x^2 y^2}[/tex]
Answer (2)
Rewrite the division as multiplication by the reciprocal: x 2 y 2 x + 1 ÷ x 2 y 2 x 2 − 1 = x 2 y 2 x + 1 ⋅ x 2 − 1 x 2 y 2 .
Cancel the common factor x 2 y 2 : x 2 y 2 x + 1 ⋅ x 2 − 1 x 2 y 2 = x 2 − 1 x + 1 .
Factor the denominator: x 2 − 1 x + 1 = ( x − 1 ) ( x + 1 ) x + 1 .
Cancel the common factor ( x + 1 ) : ( x − 1 ) ( x + 1 ) x + 1 = x − 1 1 . The final answer is x − 1 1 .
Explanation
Understanding the Problem We are asked to simplify the expression x 2 y 2 x + 1 ÷ x 2 y 2 x 2 − 1 . This involves dividing one rational expression by another, which is equivalent to multiplying by the reciprocal of the second expression. We will also need to factor and cancel common factors to simplify the expression.
Rewriting the Division First, we rewrite the division as multiplication by the reciprocal: x 2 y 2 x + 1 ÷ x 2 y 2 x 2 − 1 = x 2 y 2 x + 1 ⋅ x 2 − 1 x 2 y 2
Cancelling Common Factors Next, we cancel the common factor x 2 y 2 from the numerator and the denominator: x 2 y 2 x + 1 ⋅ x 2 − 1 x 2 y 2 = x 2 − 1 x + 1
Factoring the Denominator Now, we factor the denominator x 2 − 1 as a difference of squares: x 2 − 1 = ( x − 1 ) ( x + 1 ) So the expression becomes: ( x − 1 ) ( x + 1 ) x + 1
Cancelling Common Factors Again We can cancel the common factor ( x + 1 ) from the numerator and the denominator, provided that x e q − 1 : ( x − 1 ) ( x + 1 ) x + 1 = x − 1 1 Thus, the simplified expression is x − 1 1 .
Final Answer Therefore, the simplified expression is x − 1 1 .
Examples
Simplifying rational expressions is a fundamental skill in algebra and is used in various fields such as physics and engineering. For example, when analyzing electrical circuits, simplifying complex expressions involving impedances often requires the manipulation of rational functions. Similarly, in fluid dynamics, simplifying expressions for flow rates and pressures can involve simplifying rational expressions. These simplifications make the analysis and calculations more manageable.
To simplify the expression x 2 y 2 x + 1 ÷ x 2 y 2 x 2 − 1 , rewrite the division as multiplication by the reciprocal, then cancel the common factors, and factor the denominator. The final simplified expression is x − 1 1 .
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