Prove that 13 is an irrational number.

Let 13 is a rational number. As per the definition of rational numbers, we can express 13 = p q , {where p , q are co-primes and q ≠ 0 }. On squaring both sides, we get, 13 2 = p 2 q 2 ⇒ 13 = p 2 q 2 ⇒ p 2 = 13 q 2 . . . . . . i Which means that, p 2 is a multiple of 13 , ⇒ 13 divides p Now, let p = 13 m ⇒ p 2 = 169 m 2 Now putting this value of p 2 in equation i , 169 m 2 = 13 q 2 ⇒ 13 m 2 = q 2 ⇒ 13 divides q 2 ⇒ 13 divides q This means that, 13 is a common divisor of p and q , which contradicts the assumption done in the first step. { p , q are co-primes}. Therefore, 13 is an irrational number.

Prove that 13 is an irrational number.

Let 13 is a rational number. As per the definition of rational numbers, we can express 13 = p q , {where p , q are co-primes and q ≠ 0 }. On squaring both sides, we get, 13 2 = p 2 q 2 ⇒ 13 = p 2 q 2 ⇒ p 2 = 13 q 2 . . . . . . i Which means that, p 2 is a multiple of 13 , ⇒ 13 divides p Now, let p = 13 m ⇒ p 2 = 169 m 2 Now putting this value of p 2 in equation i , 169 m 2 = 13 q 2 ⇒ 13 m 2 = q 2 ⇒ 13 divides q 2 ⇒ 13 divides q This means that, 13 is a common divisor of p and q , which contradicts the assumption done in the first step. { p , q are co-primes}. Therefore, 13 is an irrational number.

Prove that 3 7 is an irrational number.

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

Prove that 4 3 is an irrational number.

Let us assume that 4 3 is a rational number Now, let 4 3 = a b , where a , b are co-primes and b ≠ 0 ⇒ 3 4 = b a ⇒ 3 = 4 b a Since a , b are rational numbers, then 4 b a 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, 4 3 is an irrational number.

Prove that 3 2 5 is an irrational number.

Let us assume that 3 2 5 is a rational number Now, let 3 2 5 = a b , where a , b are co-prime numbers and b ≠ 0 ⇒ 2 5 3 = b a ⇒ 5 = 3 b 2 a Since a , b are rational numbers, then 3 b 2 a is also a rational number ⇒ 5 is a rational number But, we know that, 5 is an irrational number This shows our initial assumption was incorrect Therefore, 3 2 5 is an irrational number.

Prove that 1 5 - 1 is an irrational number.

Let us assume that 1 5 - 1 is a rational number. Now, let 1 5 - 1 = a b , where a , b are co-prime numbers and b ≠ 0 ⇒ a 5 - a = b ⇒ a 5 = a + b ⇒ 5 = a + b a Since a , b are rational numbers, then a + b a is also a rational number ⇒ 5 is a rational number But, we know that, 5 is an irrational number This shows our initial assumption was incorrect. Therefore, 1 5 - 1 is an irrational number.

Prove that 1 7 - 1 is an irrational number.

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

Prove that 5 - 2 3 is an irrational number.

Let us assume that 5 - 2 3 is a rational number. Now, let 5 - 2 3 = a b , where a , b are co-prime numbers and b ≠ 0 ⇒ 2 3 = 5 - a b = 5 b - a b ⇒ 3 = 5 b - a 2 b Since a , b are integers, then 5 b - a 2 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 - 2 3 is an irrational number.

Prove that 2 - 3 is an irrational number.

Let us assume that 2 - 3 is a rational number. Now, let 2 - 3 = a b , where a , b are co-prime numbers and b ≠ 0 ⇒ 3 = 2 - a b = 2 b - a b Since a , b are integers, then 2 b - 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, 2 - 3 is an irrational number.

Prove that 3 + 2 is an irrational number.

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