Observations

The square of a number means the multiplication of the number by itself. Let us keep our discussion to two digit numbers. However idea can be extended for higher digit numbers too. For two digit number the shortcut method goes like this. Let the number be 10a+b type  where unit place digit is b and tens place digit is a then by theorem  (10a+b)^2 = 100×(a^2)+b^2+2x10xaxb = 100x (a^2)+(b^2)+ 10x(axbx2) ==> do square of a ( tens place ) , put it after two positions meant for keeping for square of b ( unit place ). To this add the multiplication of a, b, and 2 after keeping it at place leaving the first place.  This strategy solves problems fastly like, take example of 72^2 72^2, unit place 2, tens place is 7 》 square them as 4, 49 respectively,  》 keep them properly we get 4904 because 49 is to be kept leaving two places and square of unit digit is to be kept in two places format. 》 Find additional 2ab by 2x7x2= 28 》 add the 2ab to number earlier arrived at, ...

If there is a number which is perfect cube of a number, then finding its cube root is an easy job. First of all, we should know that cube root will have number of digits equal to number of triplets, whether complete triplet or partial triplet, formed from one end of that number. So if a number has 5 digits then from one end, say right, there will be one complete triplet of 3 digits, and another one partial triplet of 2 digits, so cube root will have two digits. For example cube root of 24389, will have cube root having two digits. Let us take example of finding cube root of  this number 24389. Step 1 Take last three digits 389. Consider last digit 9 of this number 389. Multiply this number 9 by itself 9, we get 81,  again multiply 81 by 9. So we get a number whose unit digit number would be 9 as 9 ✖ 9 ❌ 9 = 729. So last digit is 9. Short cut 9 x 9 = 81 ×9 = ...9 as 1 of 81 multiplying to 9 will give 9. [ Some texts mentions that for the unit digit of cube root of a perfect cu...