Question

Prove that following numbers are irrationals:
(i) $\frac{2}{\sqrt{7}}$
(ii) $\frac{3}{2\sqrt{5}}$

Answer 1

(i) SOLUTION :  

Let us assume , to the contrary ,that 2/√7 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

2/√7 = a/b

√7 a =  2b  

√7 = 2b/a

since, a & b is an integer so,2b/a  is a rational number.  

∴ √7 is rational  

But this contradicts the fact that √7 is an irrational number .

Hence, 2/√7 is an irrational .

(ii) SOLUTION :  

Let us assume , to the contrary ,that 3/2√5 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

3/2√5 = a/b

2√5 a =  3b  

√5 = 3b/2a

since, a & b is an integer so, 3b/2a

 is a rational number.  

∴ √5 is rational  

But this contradicts the fact that √5 is an irrational number .

Hence, 3+2√5 is an irrational .

HOPE THIS ANSWER WILL HELP YOU...

Answer 2

√6 is irrational

⇒√2⋅3 is irrational.

⇒√2⋅√3 is irrational

⇒√2 or √3 or both are irrational.

⇒√2+√3 is irrational.

Is this way of reasoning correc