The greatest four-digit perfect square number is

### Concept & Formula To find the greatest $n$-digit perfect square, we identify the largest number whose square is less than or equal to the largest $n$-digit number. The greatest 4-digit number is $9999$. We must find the integer root closest to it. ### Step-by-Step Solution * **Given:** We need the greatest 4-digit perfect square. * **Calculation / Deduction:** * The largest 4-digit number is $9999$. * Let's approximate its square root. We know that $100^2 = 10000$, which is a 5-digit number. * The integer just below $100$ is $99$. * Therefore, the greatest 4-digit perfect square must be $99^2$. * Calculate $99^2 = (100 - 1)^2$. * Applying the formula $(a-b)^2 = a^2 - 2ab + b^2$: $$99^2 = 10000 - 200 + 1$$ $$99^2 = 9801$$ ### Exam Strategy & Shortcut Memorize the squares of numbers up to at least $30$, and know the baseline powers of $10$ (like $100^2 = 10000$). Since $100^2$ is the smallest 5-digit square, the largest 4-digit square is immediately $(100-1)^2 = 9801$. You can solve this in 2 seconds without pen and paper. ### Common Pitfall Students often try to find the square root of $9999$ using the long division method, which consumes valuable time. Avoid long division when simple algebraic identities or baseline approximations can bridge the gap much faster. ### Final Answer **Therefore, the correct answer is 9801.**