Question

Examine, whether the following numbers are rational or irrational: 

(i)√ 7

 (ii)√ 4

 (iii) 2+√ 3 

 (iv)√ 3 +√ 2 

 (v) √3+√ 5 

 (vi) (√ 2- 2 )²

 (vii) (2-√ 2)( 2+√ 2) 

(viii) (√ 2 +√3) ²

 (ix)√ 5- 2

(x)√ 23 

(xi)√ 225 

(xii) 0.3796 

(xiii) 7.478478…… 

(xiv) 1.101001000100001……

Answer 1

√7 is not a perfect square root, so it is an irrational number.

 We have,

∴√4 can be expressed in the form of , p/ q, so it is a rational number. 

The decimal representation of √4 is 2.0.

 2 is a rational number, whereas √3 is an irrational number. 

Because, sum of a rational number and an irrational number is an irrational number, so 2 +√3  is an irrational number.

√ 2 is an irrational number. Also √3 is an irrational number. 

The sum of two irrational numbers is irrational.

 √ 3+√ 2 is an irrational number.

√ 5 is an irrational number. Also √3 is an irrational number. 

The sum of two irrational numbers is irrational.

√ 3 +√5 is an irrational number.

We have,

Now, 6 is a rational number, whereas 4√2 is an irrational number. 

The difference of a rational number and an irrational number is an irrational number. 

So, 6- 4√2  is an irrational number. 

(√2 - 2 ) ² is an irrational number.

 We have,

Since 2 is a rational number. 

(2 -√2 )(2 +√2) is a rational number.

We have,

The sum of a rational number and an irrational number is an irrational number, so 5+2√6  is an irrational number. 

(√2 +√3)² is an irrational number. 

The difference of a rational number and an irrational number is an irrational number.

 √5- 2 is an irrational number.

 √23 = 4.79583152331........ 

√225= 15 = 15/1

 Rational number as it can be represented in p/q form.

0.3796

As decimal expansion of this number is terminating, so it is a rational number. 

7.478478............ =7.bar 478 

As decimal expansion of this number is non-terminating recurring so it is a rational number.