Question

Examine, whether the following number are rational or irrational:

`(sqrt2+sqrt3)^2`

Answer 1

Let `x=(sqrt2+sqrt3)^2`be rational number

Using the formula (a + b)² = a² + b² + 2ab

`rArrx=(sqrt2)^2+(sqrt3)^2+2(sqrt2)(sqrt3)`

`rArrx=2+3+2sqrt6`

`rArrx=5+2sqrt6`

`rArr(x-5)/2=sqrt6`

`rArr(x-5)/2` is a rational number

`rArrsqrt6` is a rational number

But we know that `sqrt6` is an irrational number

So, we arrive at a contradiction

So `(sqrt2+sqrt3)^2` is an irrational number.