Question

Show that ( √3 + √5)^2 is an irrational number.​

Answer 1

Let us assume to the contrary that (√3+√5)² is a rational number,then there exists a and b co-prime integers such that,

(√3+√5)² = a/b

3+5+2√15 = a/b

8+2√15 = a/b

2√15 = a/b - 8

2√15 = (a - 8b) / b

√15 = (a - 8b) / 2b

(a - 8b) / 2b is a rational number.

Then √15 is also a rational number

But as we know √15 is an irrational number.

This is a contradiction.

This contradiction has arisen as our assumption is wrong.

Hence (√3+√5)² is an irrational number.

Hope it helps..