If possible, let √3 + √5 be a rational number equal to x. Then,
Now, x is rational
=> x2 is rational
=> x2 - 8/2 is rational
=> √15 is rational
But, √15 is rational
Thus, we arrive at a contradiction. So, our supposition that √3 +√5 is rational is wrong.
Hence, √3 + √5 is an irrational number.