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

### Concept & Formula This expression is an algebraic simplification puzzle where the goal is to eliminate the irrational components in the denominators through **rationalization**. We multiply the numerator and denominator by the conjugate to remove the surd from the bottom: $$ \frac{1}{a + \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b} $$ ### Step-by-Step Solution * **Calculation / Deduction:** * Keep the first two terms ($2 + \sqrt{2}$) as they are for now. Let's rationalize the two fractional terms individually. * **Rationalizing the Third Term:** $$ \frac{1}{2 + \sqrt{2}} $$ * Multiply by the conjugate $(2 - \sqrt{2})$: $$ = \frac{1 \times (2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})} $$ $$ = \frac{2 - \sqrt{2}}{2^2 - (\sqrt{2})^2} $$ $$ = \frac{2 - \sqrt{2}}{4 - 2} $$ $$ = \frac{2 - \sqrt{2}}{2} $$ * Break this into separate parts: $$ = \frac{2}{2} - \frac{\sqrt{2}}{2} = 1 - \frac{\sqrt{2}}{2} $$ * **Rationalizing the Fourth Term:** $$ \frac{1}{\sqrt{2} - 2} $$ * Multiply by the conjugate $(\sqrt{2} + 2)$: $$ = \frac{1 \times (\sqrt{2} + 2)}{(\sqrt{2} - 2)(\sqrt{2} + 2)} $$ $$ = \frac{\sqrt{2} + 2}{(\sqrt{2})^2 - 2^2} $$ $$ = \frac{\sqrt{2} + 2}{2 - 4} $$ $$ = \frac{\sqrt{2} + 2}{-2} $$ * Break this into separate parts and distribute the negative: $$ = -\frac{\sqrt{2}}{2} - \frac{2}{2} = -\frac{\sqrt{2}}{2} - 1 $$ * **Combining All Terms:** * Substitute the simplified fractional parts back into the main expression: $$ (2 + \sqrt{2}) + \left(1 - \frac{\sqrt{2}}{2}\right) + \left(-\frac{\sqrt{2}}{2} - 1\right) $$ * Group the integers and the surds together: $$ = (2 + 1 - 1) + \left( \sqrt{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \right) $$ * Notice that $- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = -\frac{2\sqrt{2}}{2} = -\sqrt{2}$. $$ = 2 + (\sqrt{2} - \sqrt{2}) $$ $$ = 2 + 0 = 2 $$ ### Exam Strategy & Shortcut An alternative structural approach is to look at the last term: $\frac{1}{\sqrt{2} - 2}$. You can extract a negative sign from the denominator to make it look like the third term's denominator: $\frac{1}{-(2 - \sqrt{2})} = -\frac{1}{2 - \sqrt{2}}$. Then, you have $\frac{1}{2 + \sqrt{2}} - \frac{1}{2 - \sqrt{2}}$. Finding a common denominator for this pair is highly efficient and directly yields $-\sqrt{2}$, which cleanly cancels out the $+\sqrt{2}$ at the start of the expression. ### Common Pitfall When students see $\frac{1}{\sqrt{2} - 2}$, they often incorrectly assume the conjugate is just $2 - \sqrt{2}$ without paying attention to the signs. The conjugate of $a - b$ is $a + b$. Therefore, the conjugate of $\sqrt{2} - 2$ is $\sqrt{2} + 2$. Getting the conjugate wrong will ruin the difference of squares in the denominator and break the entire solution path. ### Final Answer **Therefore, the correct answer is 2.**