$$(\sqrt{2} + \frac{1}{\sqrt{2}})^2$$ is equal to

### Concept & Formula To expand a binomial expression squared, use the standard algebraic identity: $$(a + b)^2 = a^2 + 2ab + b^2$$ ### Step-by-Step Solution * **Given:** The expression $$(\sqrt{2} + \frac{1}{\sqrt{2}})^2$$ * **Calculation:** Here, $a = \sqrt{2}$ and $b = \frac{1}{\sqrt{2}}$. Apply the formula: $$(\sqrt{2})^2 + 2(\sqrt{2})(\frac{1}{\sqrt{2}}) + (\frac{1}{\sqrt{2}})^2$$ Simplify each term individually: * $(\sqrt{2})^2 = 2$ * $2(\sqrt{2})(\frac{1}{\sqrt{2}}) = 2 \times 1 = 2$ * $(\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$ Add them together: $$2 + 2 + \frac{1}{2} = 4 + \frac{1}{2} = 4\frac{1}{2}$$ ### Exam Strategy & Shortcut Simplify the expression inside the parentheses first by finding a common denominator: $$\sqrt{2} + \frac{1}{\sqrt{2}} = \frac{2 + 1}{\sqrt{2}} = \frac{3}{\sqrt{2}}$$ Now, simply square the result directly: $$(\frac{3}{\sqrt{2}})^2 = \frac{9}{2} = 4.5 = 4\frac{1}{2}$$ Both methods are equally fast, pick the one that prevents careless errors. ### Common Pitfall * **Mistake:** Distributing the square to both terms independently, computing $(\sqrt{2})^2 + (\frac{1}{\sqrt{2}})^2 = 2 + 0.5 = 2.5$. * **Correction:** Never forget the middle term ($2ab$) when expanding a squared binomial! $(a+b)^2 \neq a^2 + b^2$. ### Final Answer **Therefore, the correct answer is 4 1/2.**