The square root of 41209 is equal to

### Concept & Strategy Determine the square root by analyzing the unit digit and using algebraic expansion to verify the closest base number. The algebraic identity used for verification: $$ (a + b)^2 = a^2 + 2ab + b^2 $$ ### Step-by-Step Solution - The given number is $41209$. - The unit digit is $9$. A perfect square ending in $9$ must have a square root ending in either $3$ or $7$ (since $3^2 = 9$ and $7^2 = 49$). - Looking at the given choices, all options end in $3$. - Next, establish the base range by squaring hundreds: $100^2 = 10000$ $200^2 = 40000$ $300^2 = 90000$ - The number $41209$ is very close to $40000$, which means its square root must be slightly greater than $200$. - Verify $203$ using the algebraic expansion: $203^2 = (200 + 3)^2$ $203^2 = 200^2 + 2(200)(3) + 3^2$ $203^2 = 40000 + 1200 + 9 = 41209$. ### Exam Strategy & Shortcut Immediately recognize that $41209$ is just slightly above $40000$ (which is $200^2$). The only option slightly above $200$ is $203$. You can select this option without even writing down a single calculation. ### Common Pitfall Choosing a number just based on the last digit without checking the magnitude. A student might guess $303$ if they incorrectly estimate $300^2$ as $30000$ instead of $90000$. ### Final Answer Therefore, the correct answer is 203.