$\sqrt{176 + \sqrt{2401}}$ is equal to

### Concept & Strategy Simplify the nested square root by finding the value of the inner radical first, adding it to the outer term, and then taking the square root of the sum. Core logic: $$ \text{Find inner root first: } \sqrt{a + \sqrt{b}} $$ ### Step-by-Step Solution - **Step 1: Evaluate the inner root, $\sqrt{2401}$.** - The number ends in $1$, so its root ends in $1$ or $9$. - Bounding ranges: $40^2 = 1600$, $50^2 = 2500$. - Since $2401$ is very close to $2500$, the root must be $49$. - **Step 2: Add to the outer term.** - Substitute $49$ back into the expression: $176 + 49$. - $176 + 49 = 225$. - **Step 3: Evaluate the final root.** - We are left with $\sqrt{225}$. - The square root of $225$ is a standard known perfect square, which is $15$. ### Exam Strategy & Shortcut Look at the outer shell: $\sqrt{176 + \text{value}}$. The nearest perfect squares above $176$ are $196$ (root $14$) and $225$ (root $15$). The inner value is $\sqrt{2401}$, which is approximately $\sqrt{2500} = 50$. Adding roughly $50$ to $176$ gives roughly $226$. The perfect square near $226$ is $225$. Thus, the answer is $\sqrt{225} = 15$. ### Common Pitfall Miscalculating the square root of $2401$. Students might guess $41$ or $51$ if they don't carefully bound the number using $40^2$ and $50^2$. ### Final Answer Therefore, the correct answer is 15.