The number of digits in the square root of 625685746009 is

### Concept & Strategy To find the number of digits in the square root of a perfect square, you do not need to calculate the actual root. You only need to count the digits of the original number and apply the pairing rule. Digit counting formula for square roots: $$ \text{Digits in root} = \lceil \frac{n}{2} \rceil $$ where $n$ is the total number of digits in the perfect square. ### Step-by-Step Solution - First, count the total number of digits in the given number $625685746009$. - The number has exactly $12$ digits. - To find the number of digits in its square root, we group the digits of the original number into pairs, starting from the rightmost digit (the unit place) and moving left. - For a $12$-digit number, the number of pairs is exactly $12 / 2 = 6$. - Each pair corresponds to one digit in the square root. - Therefore, the square root will have $6$ digits. ### Exam Strategy & Shortcut Simply count the digits and divide by $2$. If the number has an odd number of digits (say, $n$), add $1$ and then divide by $2$ (i.e., $(n+1)/2$). Here, $n=12$ (even), so $12/2 = 6$. ### Common Pitfall Attempting to actually estimate or calculate the square root of this massive number. This wastes an immense amount of time. Always read the question carefully; it asks for the "number of digits", not the root itself. ### Final Answer Therefore, the correct answer is 6.