If there is plus sign between root3 and root 5..
Let √3+√5 be a rational number.
A rational number can be written in the form of p/q where p,q are integers.
√3+√5 = p/q
√3 = p/q-√5

Squaring on both sides,
(√3)² = (p/q-√5)²
3 = p²/q²+√5²-2(p/q)(√5)
√5×2p/q = p²/q²+5-3
√5 = (p²+2q²)/q² × q/2p
√5 = (p²+2q²)/2pq

p,q are integers then (p²+2q²)/2pq is a rational number.
Then √5 is also a rational number.
But this contradicts the fact that √5 is an irrational number.
So,our supposition is false.
Therefore, √3+√5 is an irrational ...

Introduction
To prove that √5 √3 is an irrational number, we need to show that it cannot be expressed as a ratio of two integers.

Assumption
Assume that √5 √3 is a rational number and can be expressed as a ratio of two integers.

Derivation
Let √5 √3 = p/q, where p and q are integers that have no common factors.
Squaring both sides, we get:
15 = p^2/q^2

Multiplying both sides by q^2, we get:
15q^2 = p^2

This means that p^2 is a multiple of 15.
Since 15 is a product of primes, p^2 must contain at least one factor of 5 and one factor of 3.

Conclusion
This implies that p must be a multiple of √5 √3, which contradicts our assumption that p and q have no common factors.

Final Statement
Therefore, √5 √3 cannot be expressed as a ratio of two integers, and is therefore an irrational number.