 Hi and welcome to the session, I am Asha and I am going to help you with all the following question which says evaluate the following products without multiplying directly. So first let us learn a simple identity which says the product of two binomials suppose the first binomial is x plus a, the second one is x plus b then their product is equal to x square plus a plus b into x plus a into b. So this identity is a key idea that we will be using in this problem to find the following products. Let us now start with the solution. The first one is 103 into 107. Now 103 in the written as 100 plus 3 into 107 can be written as 100 plus 7. Now when comparing it with the left hand side of the identity we find here that x is equal to 100, a is equal to 3 and b is equal to 7. So in applying this identity first we have x square that is 100 square plus a plus b that is 3 plus 7 into x which is 100 plus a into b, a is 3 and b is 7. Now in simplifying it further 100 square is 10000, 7 plus 3 is 10 into 100 plus 7 into 3 is 21. So we have 10000 plus 1000 plus 21. Now in adding 10000 plus 1000 and 21 we get 110021 and hence the product of 103 and 107 by using the above identity is 110021. So this completes the first part. Let us now proceed on to the second part which is 95 into 96. Now 95 can be written as 100 minus 5 and 96 can be written as 100 minus 4 and on comparing it with the key idea we find that here x is equal to 100, a is equal to minus 5 and b is equal to minus 4. So applying the key idea we have 100 square plus in the bracket we have a plus b that is minus 5 plus minus 4 into x which is 100 plus a into b, a is minus 5 and b is minus 4. Now 100 square is 10000 plus minus 5 plus minus 4 is minus 9 into 100 plus 5 into 4 is 20 with a positive sign since minus into minus is plus. Now in adding 10,000 with 20 we get 10,020 plus on multiplying minus 9 with 100 we get minus 900. So this is further equal to 10,020 minus 900 and on subtracting 900 from 10,020 we get 9,120 hence the product of 95 into 96 by using the identity of a key idea is 91120. So this completes the second part and now proceeding on to the last one here we have to find the product of 104 and 96. Now 104 can be written as 100 plus 4 and 96 can be written as 100 minus 4 and on comparing it with our key idea we find here that x is equal to 100, a is equal to 4 and b is equal to minus 4 and on applying the identity we have 100 square plus a plus b, a is 4 plus minus 4 into x which is 100 plus a into b so 4 into minus 4. Now 100 square is 10000 plus 4 plus minus 4 is 0 into 100 plus 4 into minus 4 is minus 16 which is further equal to 10,000 plus 0 into 100 is 0 plus into minus is minus so we have minus 16 and on subtracting 16 is equal to 9984 and that is the product of 104 into 96 by using the suitable identity is 9984. So this completes the third part and hence the solution so hope you enjoyed this session take care and have a good day.