Find the greatest four-digit number, which is a perfect square, and also find its square root:

Hint: As we have been asked to find the greatest 4 – digit number which is a perfect square, we will find the smallest 5 – digit number which is a perfect square, and then a number less than this number will be the desired answer for this question.

Complete step-by-step solution
For finding the smallest 5 – digit number we first have to use some hit and try the method,
The smallest five-digit number will be 10000 and the greatest 4 – digit number will be 9999.
Let’s look at ${{50}^{2}}=2500$ which is very less than the greatest 4 digit number.
Now let’s see ${{90}^{2}}=8100$, we are getting close to the desired number.
Now let’s check if our greatest 4 – digit number or smallest five-digit number is a perfect square or not.
Now if we look closely 10000 = ${{100}^{2}}$ ,
Hence it is a perfect square and it is clear that our number will be one less than 100.
So, our desired answer is ${{99}^{2}}=9801$,
Hence, our desired 4 – digit number with a perfect square is 9801 and its square root is 99.

Note: One might get confused that how do we know 4 – digit largest number and for the smallest 5 – digit we just need to add 1 to the largest 4 – digit. Now we know that from 0 – 9 the largest is 9 largest number and if we have 4 places to fill the number and we want to make it largest then the only answer is by filling all the four places with 9 and hence 9999 is the largest 4 – digit number.