Question

prove that root 2+root 3 is irrational

Answer 1

Hey there!!!

Thank u for your Question !

Suppose 
$2 + \sqrt{3}$
is rational, say  r so that

$r \: = 2 + \sqrt{3}$


Squaring both sides, we have 
$2 + 2 \sqrt{6} + 3 = \: {r}^{2}$
which means that ,
$\sqrt{6} = {r}^{2} - 5$


Since the set of rational numbers is closed under multiplication and addition,.
${r}^{2} - 5$
 is therefore rational. However, as we know /have proved earlier that √6  is an irrational. A contradiction!

Therefore, 2 +√3 is irrational.

Hope this helps u!!

Answer 2

√2+√3is an irrational number
√2is irrational by p/qform
√3is irrational by p/q form
let √2+√3be a rational
√2+√3=p/q
square on both sides
2+2√6+3=p/q
by this √2+√3 is irrational number hence proved