 Welcome to the session. I am Asha and I am going to help you solve the following problem which says factorize each of the following. So first let us learn two identities with the help of which we will factorize both these problems. First is a cube plus b cube is equal to a plus b into a square minus a b plus b square and the second is a cube minus b cube is equal to a minus b into a square plus a b plus b square. So these two identities are a key idea with the help of which we will factorize the two given problems. Let us now start with the solution and the first one we have to factorize is 27 y cube plus 125 z cube. Now 27 y cube can be written as 3 y whole cube and 125 z cube can be written as 5 z whole cube. Now this is in the form of a cube plus b cube where a is equal to 3 y and b is equal to 5 z and that is when applying identity 1 this can sort of be written as a plus b that is 3 y plus 5 z and the next bracket we have a square that is 3 y whole square minus a into b that is 3 y into 5 z plus b square which is 5 z whole square is further equal to 3 y plus 5 z 3 y whole square is 9 y square and on multiplying 3 y with 5 z gives minus 15 y z and 5 z whole square is 25 z square. Hence on factorizing 27 y cube plus 125 z cube we get 3 y plus 5 into 9 y square minus 15 y z plus 25 z square. So this completes the first part and now proceeding on to the second part where we have to factorize 64 m cube minus 343 n cube. Now 64 m cube can be written as 4 m whole cube minus 343 n cube can be written as 7 n whole cube. Again this is in the form of a cube minus b cube and here a is equal to 4 m and b is equal to 7 n does not apply the second identity it can be written as 4 m minus 7 n and the next bracket a square that is 4 m whole square plus a into b that is 4 m into 7 n plus b square that is 7 n whole square which can further be written as 4 m minus 7 n. In the next bracket 4 m whole square is 16 m square 4 m into 7 n is 28 m n plus 7 n whole square is 49 n square. Thus on factorizing 64 n cube minus 343 n cube we get the factors as 7 n minus 7 n into 16 m square plus 28 m n plus 49 n square. So this completes the second part and hence the session hope you enjoyed it take care and have a good day.