 Hi and welcome to the session I am Shashi and I am going to help you to solve the following question. Question is proof that root 5 is irrational. Let us now start with the solution. Let us assume root 5 is rational root 5 equal to a upon b where a and b are coprime such that b is not equal to 0. Let us recall that two numbers are coprime if they have no common factor other than 1. Now root 5 is equal to a upon b which implies b root 5 is equal to a. Now squaring both sides we get 5 b square is equal to a square. Clearly we can see 5 and b square both are factors of a square. So we can write a square is divisible by 5. This further implies a is divisible by 5 because if prime number b divides a square then b divides a where a is any positive integer. So we can write a is equal to 5 c where c is any integer. Let us name the equation obtained above as 1 that is 5 b square is equal to a square We name this equation as 1. Now substituting the value of a in equation 1 we get b square is equal to 5 c whole square. This implies 5 b square is equal to 25 c square. This further implies b square is equal to 5 c square. Now clearly we can see 5 and c square both are factors of b square. So we can write b square is divisible by 5. This implies b is divisible by 5. This is because if prime number b divides a square then b divides a where a is any positive integer. Since a is divisible by 5 and b is also divisible by 5. So therefore a and b have at least 5 as common factor. But this contradicts the fact that a and b are co-prime. So our assumption that root 5 is rational is incorrect. Therefore we conclude root 5 is irrational. This is our required answer. This completes the session. Hope you understood the session. Take care and goodbye.