I face this problem too so I memorized some of the squares of the whole numbers. Such as 282=784. But I suddenly found out that using calculator is better because......
Firstly, it is impossible to memorize sqrt of all the numbers. Numbers are infinite.
Secondly, There are fractions and decimals, I can't rely on my memory for that.
Thirdly, there are extremely big and extremely small numbers.
So after all, using calculator is the best.