Question

If p, q are prime positive integers, prove that √p + √q is an irrational number.

Answer 1

Since it is given that p and q are prime positive integer.

Let us assume that √p + √q is a rational number of the form \(\frac{a}{b}\),

Squaring both sides we get,

Apply the formula (a - b)² = a² + b² - 2ab

Since p, q are integers therefore \(\cfrac{a^2-pb^2-qb^2}{ab}\) is rational number

But we know √p is irrational number,

Therefore it is a contradiction.

Hence √p - √q is irrational