Simplify.
$\frac{\frac{1}{4}-\frac{1}{y^2}}{\frac{1}{2}+\frac{1}{y}}$
Answer (2)
Factor the numerator as a difference of squares: 4 1 − y 2 1 = ( 2 1 − y 1 ) ( 2 1 + y 1 ) .
Rewrite the expression as 2 1 + y 1 ( 2 1 − y 1 ) ( 2 1 + y 1 ) .
Cancel the common factor of ( 2 1 + y 1 ) from the numerator and the denominator.
Simplify the remaining expression 2 1 − y 1 by finding a common denominator, which gives 2 y y − 2 .
The final simplified expression is 2 y y − 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 1 + y 1 4 1 − y 2 1 .
Factoring the Numerator First, we recognize that the numerator is a difference of squares: 4 1 − y 2 1 = ( 2 1 ) 2 − ( y 1 ) 2 . We can factor it as ( 2 1 − y 1 ) ( 2 1 + y 1 ) .
Rewriting the Expression Now we rewrite the original expression using the factored form of the numerator: 2 1 + y 1 4 1 − y 2 1 = 2 1 + y 1 ( 2 1 − y 1 ) ( 2 1 + y 1 ) .
Canceling Common Factors We can cancel the common factor of ( 2 1 + y 1 ) from the numerator and the denominator, provided that 2 1 + y 1 = 0 , which means y = − 2 . This gives us: 2 1 + y 1 ( 2 1 − y 1 ) ( 2 1 + y 1 ) = 2 1 − y 1 .
Combining Fractions To simplify further, we find a common denominator for the two fractions: 2 1 − y 1 = 2 y y − 2 y 2 = 2 y y − 2 .
Final Answer Therefore, the simplified expression is 2 y y − 2 .
Examples
Rational expressions are useful in many fields, such as physics and engineering, where they can be used to model complex relationships between variables. For example, in electrical engineering, rational expressions can be used to describe the impedance of a circuit as a function of frequency. Simplifying these expressions allows engineers to analyze and design circuits more efficiently. In everyday life, understanding rational expressions can help in optimizing resource allocation, such as calculating the most efficient way to distribute costs or materials.
The simplified expression is 2 y y − 2 . This was achieved by factoring the numerator, canceling a common term, and finding a common denominator. When simplified, the expression reveals a clearer relationship between the variables.
;
