$\sqrt{2 + \sqrt{3}} \cdot \sqrt{2 + \sqrt{2 + \sqrt{3}}} \cdot \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}} \cdot \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{3}}}}$ is equal to

### Concept & Formula This problem uses the algebraic identity for the difference of two squares applied continuously from right to left under nested square roots. $$(a+b)(a-b) = a^2 - b^2$$ When numbers are multiplied under square roots of the same degree, we can combine them: $$\sqrt{x} \cdot \sqrt{y} = \sqrt{x \cdot y}$$ ### Step-by-Step Solution * **Calculation / Deduction:** * Focus on the last two terms on the right: $$\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}} \cdot \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{3}}}}$$ * Combine them under a single square root. Let $a = 2$ and $b = \sqrt{2 + \sqrt{2 + \sqrt{3}}}$. * This forms $\sqrt{(a+b)(a-b)} = \sqrt{a^2 - b^2}$. $$= \sqrt{2^2 - (\sqrt{2 + \sqrt{2 + \sqrt{3}}})^2}$$ $$= \sqrt{4 - (2 + \sqrt{2 + \sqrt{3}})}$$ $$= \sqrt{2 - \sqrt{2 + \sqrt{3}}}$$ * Now, multiply this result by the second term in the original expression: $$\sqrt{2 + \sqrt{2 + \sqrt{3}}} \cdot \sqrt{2 - \sqrt{2 + \sqrt{3}}}$$ * Apply the same logic. Let $a = 2$ and $b = \sqrt{2 + \sqrt{3}}$. $$= \sqrt{2^2 - (\sqrt{2 + \sqrt{3}})^2}$$ $$= \sqrt{4 - (2 + \sqrt{3})}$$ $$= \sqrt{2 - \sqrt{3}}$$ * Finally, multiply this result by the first term in the original expression: $$\sqrt{2 + \sqrt{3}} \cdot \sqrt{2 - \sqrt{3}}$$ * Let $a = 2$ and $b = \sqrt{3}$. $$= \sqrt{2^2 - (\sqrt{3})^2}$$ $$= \sqrt{4 - 3}$$ $$= \sqrt{1} = 1$$ ### Exam Strategy & Shortcut For infinitely or finitely nested square roots of the form $(2 + \sqrt{\dots}) \cdot (2 - \sqrt{\dots})$, the pattern collapses rapidly inward. Each step eliminates one layer of the root. Recognize the $(x+y)(x-y)$ pattern immediately and collapse it backwards mentally to save writing time. ### Common Pitfall Getting intimidated by the length of the expression and attempting to calculate inner roots first. Always look for structural algebraic patterns (like difference of squares) before attempting brute-force calculation. ### Final Answer **Therefore, the correct answer is 1.**