Insert any two irrational numbers between 3 and 6 .

We are given to insert any two irrational numbers between 3 and 6 . We have, 3 2 = 3 and 6 2 = 6 . Now, we have to choose any two numbers between 3 and 6 which are not perfect squares. Arbitrarily, we are choosing 3 . 5 and 5 , such that, 3 < 3 . 5 < 5 < 6 . The square roots of these two numbers will be the required irrational numbers. Hence, 3 . 5 and 5 are irrational numbers between the irrational numbers 3 and 6 .

Insert any two irrational numbers between 3 5 and 2 3 .

We are given to insert any two irrational numbers between 3 5 and 2 3 . We have, 3 5 2 = 45 and 2 3 2 = 12 . Now, we have to choose any two numbers between 12 and 45 which are not perfect squares. Arbitrarily, we are choosing 20 and 30 , such that, 12 < 20 < 30 < 45 . The square roots of these two numbers will be the required irrational numbers. Now, 20 = 2 × 2 × 5 = 2 5 and 30 = 30 . Hence, 2 5 and 30 are irrational numbers between the irrational numbers 2 3 and 3 5 .

Write any two rational numbers between 2 and 3

We are given to write any two rational numbers between 2 and 3 . We have, 2 2 = 4 and 3 2 = 3 . Now, we have to choose any two numbers between 3 and 4 which are perfect squares. Arbitrarily, we are choosing 3 . 24 and 3 . 61 , such that, 3 < 3 . 24 < 3 . 61 < 4 . The square roots of these two numbers will be the required rational numbers. And, we have, 3 . 24 = 1 . 8 × 1 . 8 = 1 . 8 = 18 10 = 9 5 . And, 3 . 61 = 1 . 9 × 1 . 9 = 1 . 9 = 19 10 . Hence, 9 5 and 19 10 are rational numbers between the numbers 2 and 3 .

Write any two rational numbers between 7 and 8 .

We are given to write any two rational numbers between 7 and 8 . We have, 7 2 = 7 and 8 2 = 8 . Now, we have to choose any two numbers between 7 and 8 which are perfect squares. Arbitrarily, we are choosing 7 . 29 and 7 . 84 , such that, 7 < 7 . 29 < 4 . 84 < 8 . The square roots of these two numbers will be the required rational numbers. And, we have, 7 . 29 = 2 . 7 = 27 10 . And, 7 . 84 = 2 . 8 = 28 10 = 14 5 . Hence, 27 10 and 14 5 are rational numbers between the numbers 7 and 8 .

Check, whether the following is irrational or not. 80 5

If a number cannot be expressed in the form p q , where p , q ( q ≠ 0 ) are integers and do not have any common factor except 1 , then it is an irrational number. Examples of irrational numbers: 2 , 3 , 5 , π = 22 7 , . . . . . . . . etc. And, if a number can be expressed in the form p q , where p , q ( q ≠ 0 ) are integers and do not have any common factor except 1 , then it is a rational number. Examples of irrational numbers: 2 7 , 15 23 , 9 = 9 1 , 11 = 11 1 , . . . . . . etc. Now, we have, 80 5 = 2 × 2 × 2 × 2 × 5 5 = 2 × 2 × 5 5 = 4 = 4 1 . Hence, 80 5 is not an irrational number. It is a rational number.

Check, whether the following is irrational or not. ( 7 - 3 ) 2

If a number cannot be expressed in the form p q , where p , q ( q ≠ 0 ) are integers and do not have any common factor except 1 , then it is an irrational number. Examples of irrational numbers: 2 , 3 , 5 , π = 22 7 , . . . . . . . . etc. And, if a number can be expressed in the form p q , where p , q ( q ≠ 0 ) are integers and do not have any common factor except 1 , then it is a rational number. Examples of irrational numbers: 2 7 , 15 23 , 9 = 9 1 , 11 = 11 1 , . . . . . . etc. Now, we have, ( 7 - 3 ) 2 = 7 2 - 2 × 7 × 3 + 3 2 ∵ ( a - b ) 2 = a 2 - 2 a b + b 2 = 7 - 2 21 + 3 = 10 - 2 21 The number 10 - 2 21 cannot be expressed as p q . Hence, ( 7 - 3 ) 2 is an irrational number. It is not a rational number.