Question

Column A: \(|x^2|\)
Column B: \(|x|^2\)

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

Hint:

Remember this fundamental identity: the square of a number is always the same as the square of its absolute value. This is because squaring any real number, positive or negative, results in a positive outcome.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question tests the properties of absolute values and exponents for any real number \(x\).
Step 2: Detailed Explanation:
Let's analyze each column separately.
Column A: \(|x^2|\)
For any real number \(x\), the value of \(x^2\) is always non-negative (i.e., \(x^2 \ge 0\)). The absolute value of a non-negative number is the number itself.
Therefore, \(|x^2| = x^2\).
Column B: \(|x|^2\)
This expression tells us to first take the absolute value of \(x\), and then square the result. Let's test a few cases:
\begin{itemize} \item If \(x\) is positive, e.g., \(x=3\): \(|3|^2 = 3^2 = 9\). \item If \(x\) is negative, e.g., \(x=-3\): \(|-3|^2 = 3^2 = 9\). \item If \(x\) is zero, e.g., \(x=0\): \(|0|^2 = 0^2 = 0\). \end{itemize} In all cases, the result of \(|x|^2\) is the same as the result of \(x^2\). So, \(|x|^2 = x^2\).
Comparison:
From our analysis, we found that \(|x^2| = x^2\) and \(|x|^2 = x^2\).
Therefore, \(|x^2| = |x|^2\) for all real numbers \(x\).
Step 3: Final Answer:
The two quantities are equal.