Simplify the following and express your answer in positive index form.
$\frac{(a+b)^n}{b c^2} \div \frac{(a+b)^{n+3}}{a b c}$
Answer (2)
The expression b c 2 ( a + b ) n ÷ ab c ( a + b ) n + 3 simplifies to c ( a + b ) 3 a when rewritten using multiplication and applying the exponent rules.
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Rewrite the division as multiplication by the reciprocal: b c 2 ( a + b ) n × ( a + b ) n + 3 ab c .
Combine the terms into a single fraction: b c 2 ( a + b ) n + 3 ( a + b ) n ab c .
Cancel out common factors: c ( a + b ) n + 3 a ( a + b ) n .
Apply the exponent rule: c ( a + b ) 3 a .
The simplified expression with positive indices is: c ( a + b ) 3 a .
Explanation
Understanding the problem We are asked to simplify the expression b c 2 ( a + b ) n ÷ ab c ( a + b ) n + 3 and express the answer using positive indices only.
Rewrite division as multiplication First, we rewrite the division as multiplication by the reciprocal: b c 2 ( a + b ) n ÷ ab c ( a + b ) n + 3 = b c 2 ( a + b ) n × ( a + b ) n + 3 ab c .
Combine terms Next, we combine the terms into a single fraction: b c 2 ( a + b ) n × ( a + b ) n + 3 ab c = b c 2 ( a + b ) n + 3 ( a + b ) n ab c .
Cancel common factors Now, we cancel out common factors. We can cancel b from the numerator and denominator. We can also cancel c from the numerator and c 2 in the denominator, leaving c in the denominator. This gives us: b c 2 ( a + b ) n + 3 ( a + b ) n ab c = c 2 ( a + b ) n + 3 ( a + b ) n a c = c ( a + b ) n + 3 a ( a + b ) n .
Apply exponent rule Using the exponent rule x n x m = x m − n , we have: c ( a + b ) n + 3 a ( a + b ) n = c ( a + b ) n + 3 − n a = c ( a + b ) 3 a .
Final Answer The expression is now simplified and has positive indices.
Examples
This type of simplification is useful in various fields, such as physics and engineering, where complex expressions need to be simplified to make calculations easier. For example, when dealing with fluid dynamics or electrical circuits, simplifying expressions can help in understanding the relationships between different variables and making predictions about the system's behavior. It also helps in optimizing the design and performance of systems by identifying the key parameters that have the most significant impact.
