 Hello and welcome to the session. In this session, we will discuss a question which says that part a show that 1 plus omega q the whole into 1 plus omega raise to power 6 the whole into 1 plus omega raise to power 12 the whole into 1 plus omega raise to power 24 the whole up to 2 n factors is equal to 2 raise to power 2 n and part b is if x is equal to a minus b y is equal to a into alpha minus b into beta z is equal to a into beta minus b into alpha where alpha and beta are complex qubits of unity then show that x into y into z is equal to a cube minus b cube now before starting the solution of this question we should know some results and that are if the three qubits of unity are denoted by 1 omega and omega square then the sum after three qubits of unity is 0 that is 1 plus omega plus omega square is equal to 0 and certainly the product of the three qubits of unity is 1 that is 1 into omega into omega square is equal to 1 which implies omega q is equal to 1 now these results will work out as a key idea for solving out this question and now we will start with the solution firstly we will start with part a now in the part a we have to find 1 plus omega cube the whole into 1 plus omega raise to power 6 the whole into 1 plus omega raise to power 12 the whole into 1 plus omega raise to power 24 the whole up to 2 n factors now this is equal to 1 plus omega raise to power 3 the whole into 1 plus now omega raise to power 6 can be written as omega raise to power 3 whole raise to power 2 the whole into 1 plus now here omega raise to power 12 can be written as omega raise to power 3 the whole into 1 plus, now omega raise to power 24, 10 written as omega raise to power 3 the whole. Now using this rule which is given in the key idea. Now here we will put omega raise to power 3 is equal to 1. So this will be equal to plus 1 the whole into 1 plus 1 plus 3 raise to power 2 will be equal to 1 the whole into 1 plus it will be 1 raise to power 4 which is equal to 1. The whole into 1 plus now here 1 raise to power 8 which is also equal to 1. The whole into 1 factors. Now this is also 1 plus 1 is into 2 into 2 this will be equal to 3 raise to power 3 the whole into 1 plus omega raise to power 6 the whole into 1 plus omega raise to power 12 the whole into 1 plus omega raise to power 24 the whole start with the b part. Now the b part x is given to us y is also given and z is also given and beta are complex q roots of unity and we have to show this and beta x q roots of unity are given by this the sum of two groups of unity is equal to 0. Now here let alpha is equal to omega and beta is equal to omega square then beta will be equal to alpha plus omega square is equal to 0 so here alpha plus alpha square is equal to 0 is equal to minus 1. Now given a minus b into alpha minus b into beta and z is equal to a into beta minus b into alpha into z. Now this will be equal to y and z here this will be a minus b the whole minus b into beta the whole into a into beta minus b into alpha the whole. So here this will be equal to a minus b the whole into a into alpha minus b into alpha square the whole into a into alpha square minus b into alpha the whole. Now we will multiply these and this will be equal to a minus b the whole into alpha raise to power 3 minus a into b into alpha square minus a into b into alpha raise to power 4 whole into alpha raise to power 3 the whole of the q roots of unity is equal to 1. Therefore here the product of the q roots of unity will be 1 into alpha into alpha square which is equal to 1 and this implies alpha raise to power 3 is equal to 1. Now we will use this result here. Now here alpha raise to power 3 raise to power 4 can be written as b into alpha which is further equal to now here alpha raise to power 3 is 1 so it will be 1 into alpha which is equal to alpha. Now given these values here this will be equal to a minus b the whole into alpha raise to power 3 is 1 minus a into b into alpha square into b into alpha so it will be alpha raise to power 3 is again 1 equal to a minus b the whole into and from these two terms minus a b is common so it will be minus a b into this result. Alpha will be equal to minus 1 so this will be equal to a minus b the whole into minus a b minus 1 plus b square the whole which is further equal to a minus b the whole into a square plus a b plus b square the whole. Now using this formula y into z is equal to a q you should note the given question and that's all for this session hope you all have enjoyed the session.