Question

prove that root 3 + root 5 is irrational

Answer 1

Hello Mate!

$let \: x = \sqrt{3} + \sqrt{5} \\ {x}^{2} = {( \sqrt{3} + \sqrt{5} )}^{2} \\ {x}^{2} = 3 + 5 + 2 \sqrt{15} \\ \frac{ {x} ^{2} - 8 }{2} = \sqrt{15}$

Here, ( x^2 - 8 ) / 2 is rational which is equivalent to root 15 but we arrise to a contradiction that root 15 is irrational.

That is why !

$\sqrt{3} + \sqrt{5} \: is \: irrational$

Hope it helps☺!