Prove that root 3 + root 5 is an irrational no

Use proof by contradiction. Assume that the sum is rationial, that is

2-√ + 5-√ = ab 2+5 = ab

where aa and bb are integers with b≠0 b≠0. Now rewrite this as

5-√ = ab - 2-√. 5 = ab - 2.

Squaring both sides of this equation we obtain

5 = a2 b2 - 2 2-√ ab + 2. 5 = a2 b2 - 22 ab + 2.

Now, carefully solve for 2-√ 2 and obtain

2-√ = - 3b 2a + a 2b. 2 = - 3b 2a + a 2b.

This implies that 2-√ 2 is a rational number which is a contradiction. Thus

2-√ + 5-√ 2+5

is an irrational number.