Question

Examine, whether the following number are rational or irrational:

`sqrt3+sqrt2`

Answer 1

Let `x=sqrt3+sqrt2` be the rational number

Squaring on both sides, we get

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

`rArrx^2=3+2+2sqrt6`

`rArrx^2=5+2sqrt6`

`rArrx^2-5=2sqrt6`

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

Since, x is rational number

⇒ x² is rational

⇒ x² - 5 is rational

`rArr(x^2-5)/2` is rational number

`rArr sqrt6` is rational number

But, `sqrt6` is an irrational number

So, we arrive at contradiction

Hence, `sqrt3+sqrt2` is an irrational number.