Question

Show that the following numbers are irrational
(i) $\frac{1}{\sqrt{2} }$
(ii) $\sqrt[7]{5}$

Answer 1

(i) SOLUTION :  

Let us assume , to the contrary ,that 1/√2 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

1/√2 = a/b

1 × √2 /√2×√2 = a/b  

[On rationalising the denominator]

√2/2 = a/b

√2 = 2a/b

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

∴ √2 is rational  

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

Hence, 1/√2 is an irrational .

(ii) SOLUTION :  

Let us assume , to the contrary ,that 7√5 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

7√5 = a/b

√5 = a/7b

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

∴ √5 is rational  

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

Hence, 7√5 is irrational .

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