Prove that root 3+ root 5 is an irrational no

Use proof by contradiction. Assume that the sum is rationial, that is
2–√+5–√=ab2+5=ab

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

5–√=ab−2–√.5=ab−2.

Squaring both sides of this equation we obtain

5=a2b2−22–√ab+2.5=a2b2−22ab+2.

Now, carefully solve for 2–√2 and obtain

2–√=−3b2a+a2b.2=−3b2a+a2b.

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

2–√+5–√2+5

is an irrational number.