Evaluate $\sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}}$ .

### Concept & Strategy Just like other nested radicals, this problem requires solving from the innermost square root outwards. Pay close attention to the alternating addition and subtraction signs. Core logic: $$ \text{Work from the innermost term outwards.} $$ ### Step-by-Step Solution - **Step 1:** Evaluate the innermost root: $\sqrt{9} = 3$. - Substitute this into the expression: $\sqrt{19 - 3}$. - **Step 2:** Subtract and evaluate: $19 - 3 = 16$. - $\sqrt{16} = 4$. - Substitute this back into the next layer: $\sqrt{21 + 4}$. - **Step 3:** Add and evaluate: $21 + 4 = 25$. - $\sqrt{25} = 5$. - Substitute this into the outermost layer: $\sqrt{41 - 5}$. - **Step 4:** Final evaluation: $41 - 5 = 36$. - $\sqrt{36} = 6$. ### Exam Strategy & Shortcut Analyze the outermost layer: $\sqrt{41 - \text{something positive}}$. The result must be the square root of a perfect square strictly less than $41$. The nearest perfect square less than $41$ is $36$. The square root of $36$ is $6$. Option (c) is $6$, making it a highly confident guess without doing the inner math. ### Common Pitfall Missing the alternating minus and plus signs. Carelessly adding $19 + 3$ instead of subtracting, which completely throws off the chain of perfect squares. Always triple-check the operational signs in nested roots. ### Final Answer Therefore, the correct answer is 6.