Prove that is an irrational number.

If possible, let be a rational number equal to x. Then,

= (√3)² + (√5)² + 2 × √3 × √5

= 3 + 5 + 2√15

= 8 + 2√15

x² – 8 = 2√15

Now, x is rational

x² is rational.

is rational.

√15 is rational.

But, √15 is irrational

Thus, we arrive at a contradiction. So, our supposition that √3 + √5 is rational is wrong.

Hence, + is an irrational number.