 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem which says, without actually calculating the cubes find the value of each of the following. The first one is minus 12-hole cube plus 7-hole cube plus 5-hole cube. And the second one is 28-hole cube plus minus 15-hole cube plus minus 13-hole cube. So before finding the values of these two problems, let us first learn that x cube plus y cube plus z cube minus 3xyz is equal to x plus y plus z into here plus y square plus z square minus xy minus yz minus zx. And if x plus y plus z is equal to 0, that is this term is equal to 0, then we have x cube plus y cube plus z cube plus 3xyz is equal to 0 since the product of any number with 0 is 0, which further implies that x cube plus y cube plus z cube is equal to 3xyz. So this is the key idea. We are going to use in this problem to find the value of the given two problems. Let us now start with the solution and the first one is minus 12-hole cube plus 7-hole cube plus 5-hole cube. Now this is in the form of x cube plus y cube plus z cube where x is equal to minus 12, y is equal to 7 and z is equal to 5 and the sum of these three numbers that is x plus y plus z is equal to minus 12 plus 7 plus 5 which is equal to minus 12 plus 12 is further equal to 0. So this implies that x plus y plus z is equal to 0 and hence minus 12-hole cube plus 7-hole cube plus 5-hole cube must be equal to 3 into x into y into z where x is minus 12, y is 7 and z is 5 and on multiplying 3 with minus 12 7 and 5 tips minus 1, 2, 6, 0 and hence the value of the first part is minus 12, 6, 0. So this completes the first part and now proceeding on to the second part where we have to find the value of 28-hole cube plus minus 15-hole cube plus minus 13-hole cube. Again this is in the form of x cube plus y cube plus z cube where x is equal to 28, y is equal to minus 15 and z is equal to minus 13 and the sum of x plus y plus z is equal to 28 plus minus 15 plus minus 13 and the sum of minus 15 and minus 13 is minus 28 and the sum of a number with its additive inverse is 0 so we have x plus y plus z is equal to 0 this implies x cube plus y cube plus z cube is equal to 3 x y z 28-hole cube plus minus 15-hole cube plus minus 13-hole cube is equal to 3 into 28 into minus 15 into minus 13 and on the crime we get 1, 6, 3, 8, 0 and that's the product of the second part as 1, 6, 3, 8, 0 so this completes the second part and hence the solution so hope you enjoyed this session take care and have a good day