Question

prove that 2 root 3+root 5 is irrational

Answer 1

Let's have an assumption to reach the contradiction that  2√3+√5  is rational.

Let  x = 2√3+√5,  where x is rational.

$\displaystyle x=2\sqrt{3}+\sqrt{5} \\ \\ \\ x^2=(2\sqrt{3}+\sqrt{5})^2\\ \\ \\ x^2=12+5+4\sqrt{15}\\ \\ \\ x^2=17+4\sqrt{15} \\ \\ \\ x^2-17=4\sqrt{15} \\ \\ \\ \frac{x^2-17}{4}=\sqrt{15}$

At the last step, it seems that √15 can be written in p/q form. But √15 can't be actually because it's an irrational number. So a contradiction occurs here.

This contradiction makes our earlier assumption that  2√3+√5  is rational, wrong.

Hence proved that  2√3+√5  is irrational!