Question

Column A: \(p^4 - p^6\) 
Column B: \(p^3 - p^5\) 
 

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given

Hint:

When comparing powers of a positive fraction (a number between 0 and 1), remember that higher powers result in smaller values. For example, \((1/2)^3 = 1/8\) is greater than \((1/2)^4 = 1/16\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem compares two algebraic expressions involving a variable \(p\) which is a positive fraction. The key is to simplify the comparison by factoring.
Step 2: Detailed Explanation:
1. Factor both expressions. - Column A: \(p^4 - p^6 = p^4(1 - p^2)\). - Column B: \(p^3 - p^5 = p^3(1 - p^2)\).
2. Simplify the comparison. - We are given \(0 \textless p \textless 1\). This means \(p^2\) is also between 0 and 1. - Therefore, the term \((1 - p^2)\) is a positive number. - Since we are comparing two quantities, we can divide both by the same positive number without changing the relationship. Let's divide both columns by \((1 - p^2)\). - The comparison is now between \(p^4\) and \(p^3\).
3. Compare the simplified terms. - We are given that \(p\) is a positive number (\(p\textgreater0\)). Therefore, \(p^3\) is also positive. - We can divide both quantities by \(p^3\). - The comparison is now between \(\frac{p^4}{p^3} = p\) and \(\frac{p^3}{p^3} = 1\).
4. Final Comparison. - We need to compare \(p\) and 1. - The problem states that \(0 \textless p \textless 1\). - Therefore, \(p \textless 1\). - This means the original quantity in Column A is less than the original quantity in Column B.