Question

\(a>0\)
Column A: \((4\sqrt{5a})^2\)
Column B: \(40a\)

• 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:

When squaring an expression like \(k\sqrt{m}\), a common mistake is to only square the square root term. Remember to square both the coefficient outside the radical and the radical itself: \((k\sqrt{m})^2 = k^2 \times m\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests the rules of exponents and radicals, specifically how to square a product involving a square root.
Step 2: Key Formula or Approach:
We will use the exponent rule \((xy)^n = x^n y^n\). And the property that \((\sqrt{k})^2 = k\).
Step 3: Detailed Explanation:
Let's simplify the expression in Column A:
\[ (4\sqrt{5a})^2 \]
Using the exponent rule \((xy)^2 = x^2 y^2\), we can square each part of the product separately:
\[ = (4)^2 \times (\sqrt{5a})^2 \]
Now, calculate each part:
\[ (4)^2 = 16 \]
\[ (\sqrt{5a})^2 = 5a \]
Multiply the results together:
\[ 16 \times 5a = 80a \]
So, the quantity in Column A is \(80a\).
The quantity in Column B is \(40a\).
We are given that \(a>0\). Since \(a\) is a positive number, we can compare \(80a\) and \(40a\).
Step 4: Final Answer:
Since \(80>40\) and \(a>0\), it follows that \(80a>40a\). Therefore, the quantity in Column A is greater.