 Hello folks welcome again to another session on problem-solving. I hope to so far the journey has been good and you have understood quite a lot concepts in Mathematics in especially in exponents and other such topics now to understand the concepts better We know that we have to solve a lot of problems. Yes, it is very much needed However much you hate it, but then Solving more number of problems will always Make your concepts much clearer. Okay, so let us take up another question In this question, it is says it is said x y z are positive real numbers then show that This this expression has to be proven to be equal to 1 Now how to go about it? Let us take the LHS first Okay left hand side. So we will rewrite the entire expression once again x minus 1 y And this is y to the power minus 1 z And z to the power minus 1 x. Okay Now the same thing can be written as x to the power minus 1 y whole to the power 1 by 2 Why is it? Because under root a can be written as a to the power 1 by 2. Isn't it next is y to the power minus 1 z to the power half And z to the power minus 1 x to the power half Okay, now what we can separate powers on both the variables in each term. So minus 1 To the power half times y to the power half and what rule I'm using actually so a b To the power m is equal to a to the power m times b to the power m This is what is the rule now the next term Is y to the power minus 1 whole to the power half Then z to the power half Then again z to the power minus 1 whole to the power half Into x half Okay, so now what? This is if you let let's now club all the powers of x y and z So this is x to the power 1 minus 1 by 2 y Because if you see this expression here x to the power minus 1 to the power half is nothing but x to the power minus 1 by 2 Right, so I'm writing it directly and then we'll Take this x here. So this is x to the power half again Then this y will come here y to the power half And y to the power minus half is next Then z to the power half and z to the power minus half again now This step will be x to the power minus half plus half why Because a to the power m Into b to the power m is sorry a to the power n Into b to the power m b to the power sorry a to the power m into a to the power n is a to the power m plus n Okay Now y similarly it is minus half plus half And then z similarly minus half plus half So now you simplify you'll get x to the power zero Into y to the power zero into z to the power zero And we know that anything to the power zero if it is a positive real number it will be one One into one into one It was given that x and y and z are all positive real numbers So if I raise those two zero it will be one right So hence The value is one which is equal to the rhs hence Proof right so in this problem we have to prove This expression in lhs to this expression in rhs. So let us start right now if I take lhs What is lhs if you see Lhs is a to the power minus one can be written as one upon a right divided by one upon a Plus one upon b and why is this a to the power minus n is given as one upon a to the power n correct similarly on the the second term in the Lhs is one upon a divided by one upon a minus one upon b Now it is just a matter of taking lcm n simplifying So hence the number is here it will be one by a And if you see in the denominator the lcm will be ab And hence the terms will be b and then this is a Right and then plus one upon one upon a divided by divided by ab and this is b minus a What is it now? So this is equal to nothing but one by a Into ab by a plus b isn't it and this is plus one by a into ab by b minus a Right, so if a is not equal to zero so this will go and here also a will be can a will be cancelled So now it is reduced to b upon a plus b Plus b upon b minus a Right, so hence if you now see if you take the lcm again so a plus b and here it will be b minus a Correct and then what is there on top you will get b times b minus a Plus b times a plus b Isn't it so what is this now? So if you see this is b square minus ab Plus b a plus b square i'm just opening the brackets over there and a plus b Times b minus a so if you see this ab and this ab will go So finally you're left with two b square divided by b square minus a square and why is that so if you notice this part is nothing but b plus a times b minus a So hence it is nothing but b square minus a square hence. This is the answer