 Hello and welcome to the session. Let us discuss the following question. It says find x plus one whole to the power six plus x minus one whole to the power six. Hence or otherwise evaluate root two plus one whole to the power six plus root two minus one whole to the power six. Now to solve this question we need to know the expansion of a plus b whole to the power n which is given by n c zero a to the power n plus n c one a to the power n minus one b plus n c two a to the power n minus two b to the power two. So on the last term is n c n b to the power n and the expansion of a minus b whole to the power n is n c zero a to the power n minus n c one a to the power n minus one b plus n c two a to the power n minus two b to the power two. So on the last term will be minus one to the power n n c n b to the power n. So this knowledge will work as key idea. Let us now proceed on with the solution. We have to obtain x plus one whole to the power six plus x minus one whole to the power six. Now we find the expansion of x plus one whole to the power six using this expansion here n is six. So this is equal to six c zero x to the power six plus six c one x to the power six minus one that is five into one here b is one six c two x to the power four into one to the power two plus six c three x to the power three into one to the power three plus six c four x to the power two into one to the power four plus c five into x into one to the power five plus six c six into one to the power six. Now we find x minus one whole to the power six using this expansion here n is six a is x and b is one. So this is equal to six c zero x to the power six minus six c one x to the power five into one plus six c two x to the power four into one to the power two minus six c three x to the power three into one to the power three plus six c four x to the power two into one to the power four minus six c five into x into one to the power five plus minus one to the power six six c six into one to the power six. Now we add these two expressions now when we add these two expressions we see that six c one x to the power five gets cancelled with minus six c one x to the power five six c three x to the power three let's cancel with minus six c three x to the power three and six c five x gets cancelled with minus six c five x. Now this expression is equal to twice of six c zero x to the power six plus twice of six c two x to the power four plus twice of six c four x to the power two plus twice of now again this is equal to taking two common we have two into six c zero x to the power six is equal to x to the power six because six c zero is one c two is fifteen so the second term is fifteen into x to the power four c four is fifteen so the third term is fifteen into x to the power two plus six c six is one so the last term is one. Hence x plus one whole to the power six plus x minus one whole to the power six is equal to two into x to the power six plus fifteen x to the power four plus fifteen x to the power two plus one. Now we have to find the value of root two plus one whole to the power six plus root to minus 1 whole to the power 6. Now here x is root 2 so we just substitute the value of x here therefore root 2 plus 1 whole to the power 6 plus root 2 minus 1 whole to the power 6 is equal to 2 into root 2 the power 6 plus 15 into root 2 to the power 4 plus 15 into root to the power 2 plus 1 again this is equal to 2 into root 2 to the power 6 is 8 and 15 into 4 is 60 15 into 2 is 30 plus 1 and this is equal to 198. Hence the answer is 2 into x to the power 6 plus 15 x to the power 4 plus 15 x to the power 2 plus 1 and to the second part answer is 198. So this completes the question hope you enjoy the session goodbye and take care.