The square root of $535.9225$ is

### Concept & Strategy To find the square root of a large decimal, use perfect squares to establish a bounding range, then use unit digit logic to pinpoint the exact answer without heavy calculation. ### Step-by-Step Solution Determine the integer bounds for the number $535.9225$. We know that $23^2 = 529$ and $24^2 = 576$. Since $535$ falls between these two, the square root must be strictly between $23$ and $24$. This immediately eliminates $24.15$ and $28.25$. Next, look at the last digit of the decimal $535.9225$, which is $5$. This means the square root must end in a $5$. Both remaining options, $23.15$ and $23.45$, end in $5$. Notice that $535$ is much closer to $529$ (which is $23^2$) than it is to $576$. Therefore, the square root will be on the lower end of the $23$ range. We can verify $23.15$ by squaring it, or by recognizing that $23.45$ would yield a number closer to the midpoint. ### Exam Strategy & Shortcut Use the range approximation method. $23^2 = 529$. The number $535$ is only slightly larger than $529$. An increase of $0.15$ makes sense, whereas an increase of $0.45$ (nearly half) would push the square much closer to $550+$. Therefore, $23.15$ is the only logical choice. ### Common Pitfall A common mistake is attempting the long division square root method on $535.9225$, which consumes far too much time under exam conditions and is prone to arithmetic errors. ### Final Answer Therefore, the correct answer is **23.15**.