The value of $\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}}$ is

### Concept & Strategy For nested square root problems, always begin evaluating from the innermost square root and work your way outwards step-by-step. Core logic: $$ \text{Simplify from inside out: } \sqrt{a + \sqrt{b}} $$ ### Step-by-Step Solution - **Step 1:** Evaluate the innermost root: $\sqrt{225} = 15$. - Substitute this back: $\sqrt{154 + 15}$. - **Step 2:** Add and evaluate the next root: $154 + 15 = 169$. - $\sqrt{169} = 13$. - Substitute this back: $\sqrt{108 + 13}$. - **Step 3:** Add and evaluate the next root: $108 + 13 = 121$. - $\sqrt{121} = 11$. - Substitute this back: $\sqrt{25 + 11}$. - **Step 4:** Add and evaluate the next root: $25 + 11 = 36$. - $\sqrt{36} = 6$. - Substitute this back into the final, outermost expression: $\sqrt{10 + 6}$. - **Step 5:** Final evaluation: $10 + 6 = 16$. - $\sqrt{16} = 4$. ### Exam Strategy & Shortcut Look at the outermost operation: $\sqrt{10 + \text{something}}$. Since the total expression evaluates to an integer (based on the options), the sum under the outermost root must be a perfect square greater than $10$. The closest perfect square greater than $10$ is $16$. If the sum is $16$, the root is $4$. This skips the entire calculation. ### Common Pitfall Trying to add the numbers while they are still under their respective square roots. You cannot add terms across different radical layers. You must resolve each radical completely before adding it to the next term. ### Final Answer Therefore, the correct answer is 4.