Question

Show that (\sqrt{3}-\sqrt{5})^2 is a rational no.

Answer 1

(√3 - √5)² is not a rational number. Its an irrational number.

I'll give you the proof for (√3-√5)² being an irrational number.

Let us assume that (√3-√5)² is a rational number which can be expressed in p/q form where p and q are integers and q≠0

(√3-√5)² = p/q

(√3)² + (√5)² - 2(√3)(√5) = p/q  (∵ using the identity (a-b)² = a² + b² - 2ab )

3 + 5 - 2√15 = p/q

8 -2√15=p/q

8 - p/q = 2√15

$\frac{8q-p}{q}$ = 2√15

$\frac{8q-p}{2q}$ = √15

Since p,q,8,2 are integers, $\frac{8q-p}{2q}$ is rational. But we know that √15 is irrational. This contradiction has arisen because of our wrong assumption that (√3-√5)² is rational.

∴ (√3-√5)² is irrational.