prove that root5 plus root3 is irrational number 

We prove that by contradiction. 
This means, we suppose √3 is rational and then get to something that contradicts this assumption. 

If √3 is rational, we can express it as a/b, a fraction in its lowest terms (i.e., gcd(a,b) = 1). Also, b is not 1, because that would mean √3 is an integer, which it obviously is not. 

Now square both sides. 
√3 = a/b 
3 = a�/b� 

since gcd(a,b) = 1, then gcd(a�,b�) = 1 (has to be). 

But from 3 = a�/b� we can express a� = 3b�. 

So a�/b� = 3b�/b� -> now this fraction is obviously NOT in its lowest terms (remember, b is not 1) -> hence the contradiction. 

=> √3 is not rational. It is irrational.

If √5 is rational, then it can be expressed by some number a/b (in lowest terms). This would mean: 
(a/b)� = 5. Squaring, 
a� / b� = 5. Multiplying by b�, 
a� = 5b�. 

If a and b are in lowest terms (as supposed), their squares would each have an even number of prime factors . 5b� has one more prime factor than b�, meaning it would have an odd number of prime factors. 

Every composite has a unique prime factorization and can 't have both an even and odd number of prime factors. This contradiction forces the supposition wrong, so √5 cannot be rational. It is, therefore, irrational.

if √5+ √3 is rational than it can be expressed in the form of a/b where b is not equal to 0 and HCF(a,b)is 1

√5+ √3 = a/b

Since a and b are integers,a/b will also be rational and √5+ √3is irrational

This contradicts the fact that√5+ √3 is irrational. Hence, our assumption that √5+ √3 is rational is false. Therefore, √5+ √3 is irrational.