Simplify: [tex]$\frac{1}{a+b}+\frac{1}{a-b}+\frac{2 b}{a^2-b^2}$[/tex]
Answer (2)
Find a common denominator for the fractions: a 2 − b 2 = ( a + b ) ( a − b ) .
Rewrite each fraction with the common denominator and combine the numerators: a 2 − b 2 ( a − b ) + ( a + b ) + 2 b .
Simplify the numerator: a 2 − b 2 2 a + 2 b = a 2 − b 2 2 ( a + b ) .
Factor the denominator and cancel the common factor ( a + b ) : ( a + b ) ( a − b ) 2 ( a + b ) = a − b 2 . The final answer is a − b 2 .
Explanation
Problem Analysis We are asked to simplify the expression a + b 1 + a − b 1 + a 2 − b 2 2 b . To do this, we will combine the fractions by finding a common denominator and simplifying the resulting expression.
Finding Common Denominator First, notice that a 2 − b 2 can be factored as ( a + b ) ( a − b ) . Thus, the common denominator for the three fractions is ( a + b ) ( a − b ) = a 2 − b 2 . We rewrite each fraction with this common denominator: a + b 1 = ( a + b ) ( a − b ) a − b = a 2 − b 2 a − b a − b 1 = ( a − b ) ( a + b ) a + b = a 2 − b 2 a + b a 2 − b 2 2 b = a 2 − b 2 2 b
Adding the Fractions Now we can add the fractions: a + b 1 + a − b 1 + a 2 − b 2 2 b = a 2 − b 2 a − b + a 2 − b 2 a + b + a 2 − b 2 2 b = a 2 − b 2 ( a − b ) + ( a + b ) + 2 b
Simplifying the Numerator Next, we simplify the numerator: a 2 − b 2 a − b + a + b + 2 b = a 2 − b 2 2 a + 2 b
Factoring Numerator and Denominator We can factor out a 2 from the numerator: a 2 − b 2 2 ( a + b ) And we can factor the denominator as a difference of squares: ( a + b ) ( a − b ) 2 ( a + b )
Canceling Common Factors Finally, we cancel the common factor of ( a + b ) from the numerator and denominator: ( a + b ) ( a − b ) 2 ( a + b ) = a − b 2 Thus, the simplified expression is a − b 2 .
Final Answer Therefore, the simplified form of the given expression is a − b 2 .
Examples
This type of simplification is useful in electrical engineering when dealing with impedances in AC circuits. If you have two impedances, Z 1 = a + b and Z 2 = a − b , and you are trying to find a combined impedance involving these terms, you might encounter an expression similar to the one we simplified. Simplifying such expressions allows engineers to more easily analyze and design circuits.
To simplify the given expression, we found a common denominator and combined the fractions. After simplifying the numerator and canceling common factors, we arrived at the final answer of a − b 2 .
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