 Hello and welcome to the session. In this session, we will discuss a question which says that part A simplifies 3 plus 3 omega minus omega square the whole into 3 minus omega plus 3 omega square the whole and B part show that 5 plus 7 omega plus 5 omega square whole is 2 by 9 is equal to 5 plus 7 omega square plus 5 omega whole is 2 by 9 is equal to 512. Now before starting the solution of this question, we should know some results. Now here if the 3 cube roots of unity are denoted by 1 omega and omega square then of the 3 cube roots of unity is 0 that is 1 plus omega plus omega square is equal to 0 and secondly the product of 3 cube roots of unity is 1 that is 1 into omega into omega square is equal to 1 and this implies omega cube 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. First of all we will start with part A. Now here it is given that we have to simplify 3 plus 3 omega minus omega square the whole into 3 minus omega plus 3 omega square the whole. Now this is equal to from these two terms 3 is common so it will be 3 into 1 plus omega the whole minus omega square the whole into and here from these two terms 3 is common so it will be 3 into 1 plus omega square the whole minus omega the whole. Now using the result which is given in the key idea here 1 plus omega will be equal to minus omega square so this will be 3 into minus omega square the whole minus omega square the whole into now here from this result 1 plus omega square will be equal to minus omega so it will be 3 into minus omega the whole minus omega the whole further this is equal to minus 3 omega square minus omega square the whole into minus 3 omega minus omega the whole and this is equal to minus 4 omega square the whole into minus 4 omega the whole which is equal to 16 omega cube. Now using this result which is given in the key idea this will be equal to 16 into omega cube is 1 so it will be into 1 which is equal to 16 and now we will start with the B part. Now in the B part first of all we will find the value of 5 plus 7 omega plus 5 omega square whole is to power 9. Now from these two terms 5 is common so this will be equal to 5 into 1 plus omega square the whole plus 7 omega whole is to power 9. Now again using this result which is given in the key idea 1 plus omega square will be equal to minus omega so it will be equal to 5 into minus omega the whole plus 7 omega whole is to power 9 which is equal to minus 5 omega plus 7 omega whole is to power 9 which is equal to 2 omega whole is to power 9 which is equal to 2 is to power 9 into omega is to power 9. Now this will be equal to now 2 is to power 9 is 512 into omega is to power 9 can be written as omega is to power 3 whole is to power 3. Now using this result which is given in the key idea this will be equal to 512 into 1 is to power 3 which is equal to 512 therefore omega plus 5 omega square whole is to power 9 is equal to 512 and now we will find the value of omega square plus 5 omega whole is to power 9. Now from these two terms 5 is common so this will be equal to 5 into 1 plus omega the whole plus 7 omega square whole is to power 9 into now this will be equal to minus omega square plus 7 omega square whole is to power 9 which is equal to minus 5 omega square whole is to power 9 is to power 9 which is equal to 2 is to power 9 into omega is to power 2 into 9 which is omega is to power 2 is to power 18 can be written as omega will be using this result this will be equal to 512 into 1 is to power 6 which is equal to 512 into 1 which is equal to 512 therefore omega square plus 5 omega plus 5 is to power 9 is equal to 5 plus 7 omega square plus 5 omega whole is to power 9 is equal to 512 you should have given question and that's all for this session hope you all have enjoyed the session