Question

If a,b are prime positive integes, prove that (√a+√b) is an irrational number

Answer 1

Solution :

Let us assume √a + √b is rational.

Let √a + √b = p/q , where p,q are

integers and q ≠ 0.

√a = p/q - √b

Square both sides of the equation,

we get

a = p²/q² - 2p√b/q + b

=> 2p√b/q = p²/q² + b - a

=> 2p√b/q = ( p² + bq² - aq² )/q²

=> √b = [(q/2p )( p² + bq² - aq² )]/q²

=> √b = ( p² + bq² - aq² )/q

Since , p,q are integers (p²+bq²-aq²)/q is

rational , so √b is rational.

This contradicts the fact that √b is

irrational .

Hence , √a - √b is irrational .

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