Divide and simplify. [tex]$\frac{a+b}{a b} \div \frac{a^2-b^2}{5 a^3 b}$[/tex]
Answer (2)
Rewrite the division as multiplication by the reciprocal: ab a + b ÷ 5 a 3 b a 2 − b 2 = ab a + b ⋅ a 2 − b 2 5 a 3 b .
Factor the difference of squares: a 2 − b 2 = ( a + b ) ( a − b ) .
Cancel common factors after substitution.
Simplify the expression: a − b 5 a 2 .
Explanation
Understanding the Problem We are asked to divide and simplify the expression ab a + b ÷ 5 a 3 b a 2 − b 2 .
Rewrite as Multiplication First, we need to rewrite the division as multiplication by the reciprocal: ab a + b ÷ 5 a 3 b a 2 − b 2 = ab a + b ⋅ a 2 − b 2 5 a 3 b .
Factor Difference of Squares Next, we factor the difference of squares in the denominator: a 2 − b 2 = ( a + b ) ( a − b ) .
Substitute Factored Form Now, substitute the factored form into the expression: ab a + b ⋅ ( a + b ) ( a − b ) 5 a 3 b .
Cancel Common Factors We can now cancel common factors. We have ( a + b ) in both the numerator and the denominator, and we also have ab in both the numerator and the denominator: a + b a + b = 1 ab ab = 1 So we cancel these factors.
Simplify the Expression After canceling the common factors, we simplify the expression: ab a + b ⋅ ( a + b ) ( a − b ) 5 a 3 b = ab ( a 2 − b 2 ) 5 a 3 b = a 2 − b 2 5 a 2 = ( a − b ) ( a + b ) 5 a 2 . Further simplification yields: a − b 5 a 2
Final Answer Therefore, the simplified expression is a − b 5 a 2 .
Examples
When simplifying expressions in physics or engineering, you often encounter rational functions that need to be divided and simplified. For example, when calculating the transfer function of a system or analyzing the behavior of electrical circuits, simplifying rational expressions is crucial for obtaining manageable and interpretable results. This algebraic manipulation allows engineers and physicists to focus on the essential parameters and understand the system's behavior more effectively.
To divide and simplify ab a + b ÷ 5 a 3 b a 2 − b 2 , first convert the division to multiplication by the reciprocal, factor the difference of squares, and then cancel common factors. The final simplified expression is a − b 5 a 2 .
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