Question

given that √2 is irrational prove that 5+3√2 is an irrational number

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 .

Answer 2

let us assume to our contradiction that 5+3√2 is rational..so it can be written as a/b where a and b are Co prime..
5+3√2=a/b
3√2=a/b-5
3√2=(a-5b) /b
√2=(a-5b) /3b)
Here a, b and 5 and c are integers
so√2 is rational
but this contradict the fact that it is irrational..this contradiction has arisen due to our wrong assumption that √2 is rational
therefore √2 is irrational..

HOPE IT WORKS