Question

prove that root2-root3 is an irrational number

Answer 1

since we know that, 
difference between 2 irrational numbers is always irrational.
therefore root2-root3 is an irrational no.

Answer 2

Suppose  is rational, say  so that

.

Squaring both sides, we have  which means that .

Since the set of rational numbers is closed under multiplication and addition,  is therefore rational. However, as we have proved in the previous post,  is irrational. A contradiction!

Therefore,  is irrational.