Question

If $x = 3 + 2\sqrt{2}$, then the value of $\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)$ is:

• A. 1
• B. 2
• C. $4\sqrt{2}$
• D. $\sqrt{3}$

Hint:

Memorize $(a+b)^2 = a^2 + b^2 + 2ab$ to identify perfect surds quickly.

Answer 1

The Correct Option is B

Concept: Recognize standard surd identity.
Step 1: Rewrite $x$.
\[ x = 3 + 2\sqrt{2} = (1 + \sqrt{2})^2 \]
Step 2: Find $\sqrt{x}$.
\[ \sqrt{x} = 1 + \sqrt{2} \]
Step 3: Find reciprocal.
\[ \frac{1}{\sqrt{x}} = \frac{1}{1+\sqrt{2}} = \sqrt{2} - 1 \]
Step 4: Compute expression.
\[ \sqrt{x} - \frac{1}{\sqrt{x}} = (1+\sqrt{2}) - (\sqrt{2}-1) = 2 \]
Hence, the value is 2.