Prove that 2 3 - 7 is an irrational.

Let us assume that 2 3 - 7 is a rational number. Now, let 2 3 - 7 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 2 3 = a b + 7 ⇒ 3 = 1 2 a b + 7 Since a , b are integers, then a b + 7 is also a rational number. ⇒ 1 2 a b + 7 is a rational number. ⇒ 3 is a rational number. But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 2 3 - 7 is an irrational number.

Prove that 2 3 - 7 is an irrational.

Let us assume that 2 3 - 7 is a rational number. Now, let 2 3 - 7 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 2 3 = a b + 7 ⇒ 3 = 1 2 a b + 7 Since a , b are integers, then a b + 7 is also a rational number. ⇒ 1 2 a b + 7 is a rational number. ⇒ 3 is a rational number. But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 2 3 - 7 is an irrational number.

Prove that 5 - 3 is an irrational number.

Let us assume that 5 - 3 is a rational number. Now, let 5 - 3 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 3 = 5 - a b . Since a , b are integers, then 5 - a b is also a rational number. ⇒ 3 is a rational number. But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 5 - 3 is an irrational number.