3. Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2

Solutions: (i) 1/√2 Let us assume 1/√2 is rational. Then we can find co-prime x and y (y ≠ 0) such that 1/√2 = x/y Rearranging, we get, √2 = y/x Since, x and y are integers, thus, √2 is a rational number, which contradicts the fact that √2 is irrational. Hence, we can conclude that 1/√2 is irrational. (ii) 7√5 Let us assume 7√5 is a rational number. Then we can find co-prime a and b (b ≠ 0) such that 7√5 = x/y Rearranging, we get, √5 = x/7y Since, x and y are integers, thus, √5 is a rational number, which contradicts the fact that √5 is irrational. Hence, we can conclude that 7√5 is irrational. (iii) 6 +√2 Let us assume 6 +√2 is a rational number. Then we can find co-primes x and y (y ≠ 0) such that 6 +√2 = x/y⋅ Rearranging, we get, √2 = (x/y) – 6 Since, x and y are integers, thus (x/y) – 6 is a rational number and therefore, √2 is rational. This contradicts the fact that √2 is an irrational number. Hence, we can conclude that 6 +√2 is irrational.