Question

Examine, whether the following number are rational or irrational:

`sqrt3+sqrt5`

Answer 1

Let `x=sqrt3+sqrt5` be the rational number

Squaring on both sides, we get

`rArrx^2=(sqrt3+sqrt5)^2`

`rArrx^2=3+5+2sqrt15`

`rArrx^2=8+2sqrt15`

`rArrx^2-8=2sqrt15`

`rArr(x^2-8)/2=sqrt15`

Now, x is rational number

⇒ x² is rational number

⇒ x² - 8 is rational number

`rArr (x^2 -8)/2` is rational number

`rArrsqrt15` is rational number

But `sqrt15` is an irrational number

So, we arrive at a contradiction

Hence `sqrt3+sqrt5` is an irrational number.