 Hello and welcome to the session. In this session, we discussed the following question that says, if a upon b is equal to c upon b is equal to e upon f, prove that a cube ce upon b cube df is equal to square root of a to the power 5 c cube e square upon square root of b to the power 5 d cube f square. Let's proceed with the solution now. We are given that a upon b is equal to c upon d is equal to e upon f and we need to prove that a cube ce upon b cube df is equal to square root of a to the power 5 c cube e square upon square root of b to the power 5 d cube f square. Now first of all we assume let a upon b equal to c upon b equal to e upon f be equal to k. This means we have a is equal to bk since a upon b is equal to k c is equal to dk since c upon d is also equal to k and e is equal to fk since e upon f is equal to k. Now first of all we take the LHS which is a cube ce upon b cube df. Now we substitute a as bk c as dk and e as fk so we get this is equal to b cube k cube into dk into fk and this whole upon b cube df. Now this b cube b cube cancels d cancels with d and f cancels with f and so we are left with k cube into k into k which is equal to k to the power of 5. So we have the LHS is equal to k to the power 5. Now we consider the RHS which is equal to this that is square root of a to the power 5 c cube e square upon square root of b to the power 5 d cube f square. Now here also we will put a as bk c as dk and e as fk so we get this is equal to square root of b to the power 5 k to the power 5 into d cube k cube into f square k square upon square root of b to the power 5 d cube f square. This is further equal to square root of k to the power 5 into k to the power 3 into k to the power 2 becomes k to the power of 10 into b to the power 5 b to the power 3 and f to the power 2 upon square root of b to the power 5 d cube f square. Now square root of k to the power 10 is taken out that is we have k to the power 10 whole to the power 1 upon 2 into square root of b to the power 5 d cube f square upon square root of b to the power 5 d cube f square. Now this cancels with this and we are left with k to the power 10 whole to the power 1 upon 2 which is equal to k to the power of 5 that is the RHS is equal to k to the power of 5 and we know that LHS is also equal to k to the power of 5. So we have the LHS is equal to the RHS hence we have proved that a cube ce upon b cube df is equal to square root of a to the power 5 c cube e square upon square root of b to the power 5 d cube f square. So we have proved this so this completes the session hope you have understood the solution of this question.