Q. Prove that √2+√7 is an irrational.
Sol:
Let, √2+√7 is a rational number.
i.e., √2+√7=x, where x is an integer.
=> √2 = x-√7
=> (√2)²=(x-√7)² (squaring both sides)
=> 4 =x²+7 - 2√7x
=> 2√7x=x²+7-4
=> √7 = (x²+3)/2x
As x is an integer, so right hand side represents a rational number.
But it contradict the fact that √3 irrational. (So our assumption was wrong)
Hence √2+√7 is irrational.
