prove that under root 3 + under root 5 is irrational

Question

Prove that root 3 plus root 5 is irrational

Answer 1

Solution: Let √3+√5 be any rational number x

x=√3+√5

squaring both sides

x²=(√3+√5)²

x²=3+5+2√15

x²=8+2√15

x²-8=2√15

(x²-8)/2=√15

As x is a rational number

so x² is also a rational number, 8 and 2 are rational numbers. ,

so √15 must also be a rational number as quotient of two rational numbers is rational

But this contradicts the fact that √5 is irrational

This contradiction arose because of our false assumption
so √3+√5 is not a rational number.
So √3+√5 is irrational.