Question

Column A: \((\sqrt{5} + \sqrt{5})^2\)
Column B: 20

• 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:

A common mistake is to incorrectly apply the exponent rule \((a+b)^2 = a^2 + 2ab + b^2\). While that would work (\((\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{5}) + (\sqrt{5})^2 = 5 + 2(5) + 5 = 20\)), it's much faster to combine the like terms inside the parentheses first.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question requires simplifying an expression involving square roots and exponents.
Step 2: Key Formula or Approach:
First, simplify the expression inside the parentheses. Then apply the exponent. We will use the rule \((ab)^2 = a^2 b^2\).
Step 3: Detailed Explanation:
Let's evaluate the expression in Column A.
First, simplify the terms inside the parentheses:
\[ \sqrt{5} + \sqrt{5} = 2\sqrt{5} \]
Now, square the result:
\[ (2\sqrt{5})^2 \]
Using the exponent rule \((ab)^2 = a^2 b^2\), we get:
\[ 2^2 \times (\sqrt{5})^2 \]
Calculate each part:
\[ 2^2 = 4 \]
\[ (\sqrt{5})^2 = 5 \]
Multiply the results:
\[ 4 \times 5 = 20 \]
The quantity in Column A is 20.
Comparison:
Column A: 20
Column B: 20
The two quantities are equal.
Step 4: Final Answer:
The two quantities are equal.