Question:
\(a = -219\)
Column A: \(a^7 + a^5\)
Column B: \(a^8 + a^{18}\)
\(a = -219\)
Column A: \(a^7 + a^5\)
Column B: \(a^8 + a^{18}\)
Column A: \(a^7 + a^5\)
Column B: \(a^8 + a^{18}\)
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For questions involving large numbers and exponents, focus on the properties of numbers rather than calculation. The key rules are: (negative)\textsuperscript{odd} = negative, and (negative)\textsuperscript{even} = positive. This is often all you need to solve the problem.
Updated On: Oct 1, 2025
- The quantity in Column A is greater.
- The quantity in Column B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
This problem tests the properties of exponents, specifically the sign of a negative number raised to an odd or even power. We don't need to calculate the exact values, but rather determine the sign and relative magnitude of the terms.
Step 2: Detailed Explanation:
The given value is \(a = -219\), which is a negative number.
Analyze Column A: \(a^7 + a^5\)
\begin{itemize} \item A negative number raised to an odd power results in a negative number. \item Therefore, \(a^7 = (-219)^7\) is a negative number. \item Similarly, \(a^5 = (-219)^5\) is a negative number. \item The sum of two negative numbers is a negative number. So, the quantity in Column A is negative. \end{itemize} Analyze Column B: \(a^8 + a^{18}\)
\begin{itemize} \item A negative number raised to an even power results in a positive number. \item Therefore, \(a^8 = (-219)^8\) is a positive number. \item Similarly, \(a^{18} = (-219)^{18}\) is a positive number. \item The sum of two positive numbers is a positive number. So, the quantity in Column B is positive. \end{itemize} Step 3: Final Answer:
The quantity in Column A is a negative number, and the quantity in Column B is a positive number. Any positive number is always greater than any negative number. Therefore, the quantity in Column B is greater.
This problem tests the properties of exponents, specifically the sign of a negative number raised to an odd or even power. We don't need to calculate the exact values, but rather determine the sign and relative magnitude of the terms.
Step 2: Detailed Explanation:
The given value is \(a = -219\), which is a negative number.
Analyze Column A: \(a^7 + a^5\)
\begin{itemize} \item A negative number raised to an odd power results in a negative number. \item Therefore, \(a^7 = (-219)^7\) is a negative number. \item Similarly, \(a^5 = (-219)^5\) is a negative number. \item The sum of two negative numbers is a negative number. So, the quantity in Column A is negative. \end{itemize} Analyze Column B: \(a^8 + a^{18}\)
\begin{itemize} \item A negative number raised to an even power results in a positive number. \item Therefore, \(a^8 = (-219)^8\) is a positive number. \item Similarly, \(a^{18} = (-219)^{18}\) is a positive number. \item The sum of two positive numbers is a positive number. So, the quantity in Column B is positive. \end{itemize} Step 3: Final Answer:
The quantity in Column A is a negative number, and the quantity in Column B is a positive number. Any positive number is always greater than any negative number. Therefore, the quantity in Column B is greater.
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