Show sqrt3sqrt7 is irrational Show sqrt3sqrt7 is irrationalS

Solution

Let us take 3 + 7 is rational

i.e 3 + 7 = a , Where a is rational

7 = a -  3

Now squaring on both sides

( 7 )² = ( a -  3 )²

7 = a² + ( 3 )² - 2 . a. 3

7 = a² + 3 - 2.a.3

2 . a . 3 = a² + 3 - 7

   2 . a . 3 = a² - 4

3 = ( a² - 4 ) / 2a

Which is a contradiction as right hand side is rational but left hand side is irrational

Hence,

  3 + 7 is irrational