Question

Prove that `sqrt3+sqrt5` is an irrational number.

Answer 1

Given that `sqrt3+sqrt5` is an irrational number

Now we have to prove `sqrt3+sqrt5` is an irrational number 

Let `x=sqrt3+sqrt5` is a rational

Squaring on both sides

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

`rArrx^2=(sqrt3)^2+(sqrt5)^2+2sqrt3xxsqrt5`

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

`rArrx^2=8+2sqrt15`

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

Now  x is rational

⇒ x² is rational

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

`rArr sqrt15` is rational

But, `sqrt15` is an irrational

Thus we arrive at contradiction that `sqrt3+sqrt5` is a rational which is wrong.

Hence `sqrt3+sqrt5` is an irrational.