Simplify the problem: [tex]$\frac{24 a^6 b^{10}}{12 a^2 b^5}$[/tex]
Answer (1)
Divide the coefficients: 12 24 = 2 .
Simplify the powers of a : a 2 a 6 = a 4 .
Simplify the powers of b : b 5 b 10 = b 5 .
Combine the results: The simplified expression is 2 a 4 b 5 .
Explanation
Understanding the Problem We are given the expression 12 a 2 b 5 24 a 6 b 10 . Our goal is to simplify it by dividing the coefficients and applying the rules of exponents.
Dividing the Coefficients First, we divide the coefficients: 12 24 = 2
Simplifying Powers of a Next, we simplify the powers of a using the quotient rule for exponents, which states that a n a m = a m − n : a 2 a 6 = a 6 − 2 = a 4
Simplifying Powers of b Similarly, we simplify the powers of b using the same quotient rule: b 5 b 10 = b 10 − 5 = b 5
Combining the Results Finally, we combine the results to obtain the simplified expression: 2 a 4 b 5
Final Answer Therefore, the simplified expression is 2 a 4 b 5 .
Examples
Imagine you are organizing a bookshelf. You have 24 copies of a book and want to arrange them into stacks. If each stack has 12 books, you can determine the number of stacks by dividing 24 by 12. Similarly, when simplifying algebraic expressions, you divide the coefficients and apply exponent rules to make the expression more manageable. This is useful in various fields, such as physics, engineering, and computer science, where complex equations need to be simplified for easier calculations and analysis.
