$\sqrt{44944} + \sqrt{52441} = x$

### Concept & Strategy Solve this by approximating the individual square roots using their unit digits and bounding them between known multiples of $10$. Base approximation logic: $$ \text{Find bounds } (10x)^2 < N < (10(x+1))^2 $$ ### Step-by-Step Solution - **Part 1: Find $\sqrt{44944}$** - Unit digit is $4$, so the root ends in $2$ or $8$. - Find base boundaries: $200^2 = 40000$ and $220^2 = 48400$. $210^2 = 44100$. - $44944$ is slightly larger than $44100$. - This points to the root being $212$. - **Part 2: Find $\sqrt{52441}$** - Unit digit is $1$, so the root ends in $1$ or $9$. - Find base boundaries: $220^2 = 48400$ and $230^2 = 52900$. - $52441$ is very close to $52900$. - This means it is at the higher end of the range, pointing to $229$. - **Part 3: Addition** - Sum the two calculated roots: $212 + 229 = 441$. ### Exam Strategy & Shortcut Use unit digit analysis for the final sum. The first root ends in $2$. The second root ends in $9$. The sum of the unit digits is $2 + 9 = 11$, which means the final answer must end in $1$. Among the given options, only $441$ ends in $1$. ### Common Pitfall Getting bogged down in long division for square roots. Even if you aren't entirely sure of the exact roots, estimating them as $\sim 212$ and $\sim 230$ gives a sum around $442$, making $441$ the obvious answer by approximation alone. ### Final Answer Therefore, the correct answer is 441.