Question

The value of \( \left| \sqrt{4+2\sqrt{3}} \right| - \left| \sqrt{4-2\sqrt{3}} \right| \) is:

• A. \( 1 \)
• B. \( 2 \)
• C. \( 4 \)
• D. \( 3 \)
• E. \( 5 \)

Hint:

When simplifying \( \sqrt{a - 2\sqrt{b}} \), always ensure the larger number comes first in the subtraction (e.g., \( \sqrt{3}-1 \)) to keep the root positive.

Answer 1

The Correct Option is B

Concept: To simplify nested radicals of the form \( \sqrt{a \pm 2\sqrt{b}} \), we try to express the term inside the square root as a perfect square \( (x \pm y)^2 = x^2 + y^2 \pm 2xy \).

Step 1: Simplifying the nested radicals.
For \( 4 + 2\sqrt{3} \), we need two numbers whose sum is 4 and product is 3. These are 3 and 1. \[ 4 + 2\sqrt{3} = 3 + 1 + 2\sqrt{3}= (\sqrt{3})^2 + (1)^2 + 2(\sqrt{3})(1) = (\sqrt{3} + 1)^2 \] Similarly: \[ 4 - 2\sqrt{3} = (\sqrt{3} - 1)^2 \]

Step 2: Calculating the final value.
\[ \sqrt{(\sqrt{3} + 1)^2} - \sqrt{(\sqrt{3} - 1)^2} = (\sqrt{3} + 1) - (\sqrt{3} - 1) \] \[ = \sqrt{3} + 1 - \sqrt{3} + 1 = 2 \]