Question

Prove that following numbers are irrationals:
(iii) $4+ {\sqrt{2}}$
(iv) $5 {\sqrt{2}}$

Answer 1

(iii) SOLUTION :  

Let us assume , to the contrary ,that 4 +√2 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

4 +√2 = a/b

√2 =  a/b - 4

since, a & b is an integer so, a/b - 4 is a rational number.  

∴ √2 is rational  

But this contradicts the fact that √2 is an irrational number .

Hence, 4 +√2 is an irrational .

(iv) SOLUTION :  

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

5√2 = a/b

√2 = a/5b

since, a & 7b is an integer so, a/5b  is a rational number.  

∴ √2 is rational  

But this contradicts the fact that √2 is an irrational number .

Hence, 5√2 is irrational .

HOPE THIS ANSWER WILL HELP YOU...

Answer 2

Heyya there
Here is ur answer in the attachment
Hope it may helps you