Question

Given that ✓2 is irrational prove that 5+3✓2 and 7+5✓2 is irrational

Answer 1

Sol :

Suppose 5 +3√3 is rational .

Let 5 +3√3 = r       where r is a rational.

∴  (5 +3√3)2 = r2  

∴  25+27 +30√3 = r2

∴√3 = (r2 - 52) / 30

Now , LHS = √3 is an irrational number .

RHS  (r2 - 7) / 2 is a rational number .

But rational number cannot be equal to an irrational.

∴our supposition is wrong.

∴ 5 + 3√3 is irrational .