 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem which says evaluate the following using suitable identities. So first let us learn two identities which will help us to solve the following problem. First is, whole cube of x plus y is equal to x cube plus y cube plus 3x square y plus 3xy square and the second one is whole cube of x minus y is equal to x cube minus y cube minus 3x square y plus 3xy square. So these two identities is a key idea that will help us to evaluate the given three problems. Let us now start with the solution. The first one is 99 whole cube. Now 99 can be written as 100 minus 1 whole cube and on comparing this with the left hand side of the second identity, we find here that x is equal to 100 and y is equal to 1. That is when applying the second identity. This can be written as 100 whole cube minus y cube that is 1 cube minus 3 into x square y, x is 100 that is 100 square y is 1 plus 3 into xy square so x is 100 and y is 1 square. Now 100 cube is 1 0 0 0 0 0 minus 1 minus 3 into 100 square is 1 0 0 0 0 plus 3 into 100 is 300. Now on adding the positive terms we have 1 0 0 0 3 0 0 minus 1 and on multiplying 3 to 10,000 we get 30,000 and now adding 1 0 0 0 3 0 as it is and now adding minus 1 with 30,000 glyphs minus 3 0 0 0 1 and now on subtracting 30,000 1 from 1 0 0 0 3 0 0 we get equal to 9 7 0 2 9 9 and thus 99 whole cube is equal to 9 7 0 2 9 9. So this completes the first part and now proceeding on to the second part where we have to find 102 whole cube. Now 102 can be written as 100 plus 2 whole cube and on comparing it with the first identity we have here x is equal to say x plus y whole cube is equal to x cube plus y cube plus 3 x square y plus 3 x y square and on comparing this with the left inside of the first identity we find here that x is equal to 100 and y is equal to 2 and on applying this identity we have 100 whole cube plus y cube that is 2 whole cube plus 3 a square that is 100 square into y plus 3 x y square that is 2 square. Now 100 whole cube is 1 with 6 zeros 2 whole cube is 8 plus 3 into 100 whole square is 1 with 4 zeros into 2 plus 3 into 100 and 2 whole square is 4. Now this is further equal to 1 with 6 zeros plus 8 plus 3 into 2 is 6 into 10,000 plus 3 4s are equal to 100 which is further equal to 100000 plus 8 plus 6 6 into 1 is 6 with 4 zeros 1 2 3 4 and 12 into 100 is 1200 and on adding all these we get 1061208 that is the whole cube of 102 is equal to 1061208. So this completes the second part and now proceeding on to the third part where we have to find the whole cube of 998. Now 998 can be written as 1000 minus 2 and on comparing it with the left inside of the second identity we find here that x is equal to 1000 and y is equal to 2 and that is an applying identity. First we have x cube that is 1000 whole cube and here is minus y cube that is 2 whole cube minus 3 times of x square into y so x is 1000 square into y is 2 plus 3 into x that is 1000 y square that is 2 square which is further equal to 1000 whole cube is 1 with 9 zeros minus 2 whole cube is 8 minus 3 into 1000 whole square is 1 with 6 zeros into 2 plus 3 into 1000 and 2 whole square is 4 is further equal to 1 with 9 zeros minus 8 minus 3 into 2 is 6 with 6 zeros 3 4 5 6 plus 3 4s at 12 with 3 zeros. Now on adding 1 with 9 zeros with 12000 gives 1 0 0 0 0 12 0 0 0 minus on adding 6 with 6 along with 6 zeros gives 6 0 0 0 0 0 8 and now on simplifying it further we get the result equal to 994011992 and thus the whole cube of 998 is equal to 994011992 so this completes the third part and hence the solution. So hope you enjoyed this session and please do remember the identities used in this problem. Take care and have a good day.