Question

If 'a' is a rational root p is irrational , then prove that (a+root p ) is irrational

Answer 1

Question:

If 'a' is a rational number and '√p' is an irrational number where 'p' is a rational number, then prove that 'a + √p' is an irrational number.

Solution:

Assume to reach the contradiction that 'a + √p' is rational.

Let  x = a + √p,  where x is strictly rational.

    x = a + √p

⇒  x² = (a + √p)²

⇒  x² = a² + p + 2a√p

⇒  x² - a² - p = 2a√p

⇒  (x² - a² - p) / 2a = √p

Consider the LHS of the last step.

• As we assumed earlier, 'x' is rational, then so is 'x²'.

• 'a' is a rational number, so is 'a²'.

• 'p' is rational. It's given in the question.

Hence LHS is rational. But the RHS is irrational because there's only √p in the RHS and √p is irrational.

This makes the contradiction!

Hence Proved!