 Hi and welcome to the session. I am Asha and I am going to help you with the following question that says if x plus iota y whole cube is equal to u plus iota v, then show that u upon x plus v upon y is equal to 4x square minus y square. So, solution we are given that x plus iota y whole cube is equal to u plus iota v. So, first we will try to find the values of u and v, then we will substitute it in the left hand side and show that it comes equal to the right hand side. So, let us open this x plus iota y whole cube, we have x cube plus 3 times x square into iota y plus 3x into iota y whole square plus iota y whole cube is equal to u plus iota v, since a plus v whole cube is equal to a cube plus 3a square b plus 3ab square plus b cube or it can further written as h cube plus iota 3x square y minus since iota square is minus 1, so plus into minus is minus 3x y square minus iota y cube since iota square is minus 1 and 1 iota as it is and in the right hand side we have u plus iota v. This has sense iota square is equal to minus 1. Now on the left hand side separating the real and imaginary parts we have x cube minus 3x y square plus iota times 3x square y minus y cube is equal to u plus iota v. Now on comparing the real and imaginary parts we get u is equal to x cube minus 3x y square and v is equal to 3x square y minus y cube. And now according to the question we have to show that u upon x plus v upon y is equal to 4x square minus y square. So, substituting the values of u and v on the left hand side which is u upon x plus v upon y can be written as x cube minus 3x y square upon x plus 3x square y minus y cube upon y which can further written as x into x square minus 3y square upon x plus y into 3x square minus y square upon y. Now cancelling the common multiple we have x square minus 3y square plus 3x square minus y square which is further equal to 4x square minus 4y square is further equal to 4x square minus y square. Thus we have u upon x plus v upon y is equal to 4 times of x square minus y square. Hence proved. This completes the solution. Hope you enjoyed it. Take care and have a good day.