$\sqrt{53824} = x$

### Concept & Strategy The fastest way to find the square root of a large perfect square is by observing its unit digit and estimating its range using known base squares. Base approximation logic: $$ (xy)^2 \approx x^2 \times 100 $$ ### Step-by-Step Solution - Identify the unit digit of the given number: The number $53824$ ends in $4$. - Therefore, its square root must end in either $2$ or $8$ (since $2^2 = 4$ and $8^2 = 64$). - Look at the options: All options end in $2$, so we must estimate the magnitude. - Find the nearest base squares: $200^2 = 40000$ $300^2 = 90000$ - Since $53824$ is closer to $40000$, the root is in the lower half of the $200$-$300$ range. - Let's check $230$ and $240$: $230^2 = 52900$ $240^2 = 57600$ - The number $53824$ is just slightly larger than $52900$. - The only option that fits this tight range and ends in $2$ is $232$. ### Exam Strategy & Shortcut Use the range bounding method. $20^2 = 400$, so $200^2 = 40000$. $25^2 = 625$, so $250^2 = 62500$. The number $53824$ lies between $40000$ and $62500$, which eliminates $332$. Because $53824$ is closer to $230^2$ ($52900$) than $240^2$ ($57600$), the answer must be $232$. ### Common Pitfall Students often try to calculate the square root using the traditional long division method, which wastes precious exam time. Always use options and approximation first. ### Final Answer Therefore, the correct answer is 232.