$$\left(\sqrt{3} - \frac{1}{\sqrt{3}}\right)^2$$ simplifies to

### Concept & Formula Apply the standard algebraic expansion identity: $$(a - b)^2 = a^2 - 2ab + b^2$$ Alternatively, simplify the expression inside the parentheses first before squaring. ### Step-by-Step Solution **Method 1: Direct Fraction Simplification (Recommended)** Simplify the terms inside the parentheses by finding a common denominator: $$\sqrt{3} - \frac{1}{\sqrt{3}} = \frac{(\sqrt{3} \times \sqrt{3}) - 1}{\sqrt{3}} = \frac{3 - 1}{\sqrt{3}} = \frac{2}{\sqrt{3}}$$ Now, square the simplified result: $$\left(\frac{2}{\sqrt{3}}\right)^2 = \frac{2^2}{(\sqrt{3})^2} = \frac{4}{3}$$ ### Exam Strategy & Shortcut Converting to a single fraction first is much faster and less prone to arithmetic mistakes than expanding via $(a-b)^2$. $\sqrt{3} - 1/\sqrt{3}$ instantly becomes $2/\sqrt{3}$, and squaring it yields $4/3$ directly. ### Common Pitfall When using the $(a-b)^2 = a^2 - 2ab + b^2$ expansion, students sometimes miscalculate the middle term $2ab$ or mistake the sign of $b^2$, writing it as negative instead of positive. ### Final Answer **Therefore, the correct answer is 4/3.**