Question

Examine, whether the following number are rational or irrational:

`(sqrt2-2)^2`

Answer 1

Let `x=(sqrt2-2)^2` be a rational number.

`x=(sqrt2-2)^2`

`rArrx=2+4-4sqrt2`

`rArrx=6-4sqrt2`

`rArr(x-6)/(-4)=sqrt2`

Since, x is rational number,

⇒ x – 6 is a rational nu8mber

`rArr(x-6)/(-4)` is a rational number

⇒ `sqrt2`is a rational number

But we know that `sqrt2` is an irrational number, which is a contradiction 

So `(sqrt2-2)^2` is an irrational number