\frac{a+b}{(a-2 b)^2}-\frac{1}{2 b-a}
Answer (1)
Rewrite the second term with a common denominator.
Find a common denominator for both fractions.
Combine the fractions into a single expression.
Simplify the numerator to obtain the final simplified expression: ( a − 2 b ) 2 2 a − b .
Explanation
Understanding the Problem We are asked to simplify the expression ( a − 2 b ) 2 a + b − 2 b − a 1 .
Rewriting the Second Term First, notice that 2 b − a = − ( a − 2 b ) . We can rewrite the second term to have a denominator that is similar to the first term's denominator: − 2 b − a 1 = − − ( a − 2 b ) 1 = a − 2 b 1
Finding a Common Denominator Now, we can rewrite the original expression as ( a − 2 b ) 2 a + b + a − 2 b 1 To combine these two fractions, we need a common denominator. The common denominator is ( a − 2 b ) 2 . So, we rewrite the second term with the common denominator: a − 2 b 1 = ( a − 2 b ) 2 a − 2 b
Combining the Fractions Now, we can add the two fractions: ( a − 2 b ) 2 a + b + ( a − 2 b ) 2 a − 2 b = ( a − 2 b ) 2 a + b + a − 2 b
Simplifying the Numerator Finally, we simplify the numerator: ( a − 2 b ) 2 a + b + a − 2 b = ( a − 2 b ) 2 2 a − b So, the simplified expression is ( a − 2 b ) 2 2 a − b .
Final Answer The simplified expression is ( a − 2 b ) 2 2 a − b .
Examples
Simplifying rational expressions is a fundamental skill in algebra and calculus. For example, when solving complex circuit problems in electrical engineering, you often encounter rational functions representing impedances. Simplifying these expressions allows engineers to analyze the circuit's behavior more efficiently and design components with specific characteristics. Similarly, in chemical engineering, simplifying rational expressions can help in modeling reaction rates and optimizing reactor designs.
