Prove that √3 + √5 is an irrational number.

Let √3 + √5 be a rational number, say r
Then √3 + √5 =   r 
On squaring both sides, 
     (√3 + √5)² =   r ²
 3 + 2√15 + 5 =  r ² 
       8 + 2√15 =   r ²
             2√15 =  r ² - 8
               √15 = (  r ² - 8)/2
Now (  r ² - 8)/2 is a rational number and √ 15 is an irrational number. 
Since a rational number cannot be equal to an irrational number. Our assumption that
√3 + √5 is rational is wrong.