Proof that √3 * √5 is an irrational number:

To prove that √3 * √5 is an irrational number, we need to assume the opposite and show that it leads to a contradiction. So, let's assume that √3 * √5 is a rational number.

Assumption: √3 * √5 is a rational number.

Definition of a rational number: A rational number can be expressed in the form p/q, where p and q are integers, and q is not equal to zero.

Expressing √3 * √5 as a rational number:
√3 * √5 = p/q

Squaring both sides, we get:
(√3 * √5)^2 = (p/q)^2

Simplifying the equation, we have:
(√3)^2 * (√5)^2 = p^2/q^2
3 * 5 = p^2/q^2
15 = p^2/q^2

Consequence: From the above equation, we can conclude that p^2 is divisible by 15, and therefore, p must also be divisible by 15.

Consider the prime factorization of 15:
15 = 3 * 5

Consequence: Since p is divisible by both 3 and 5, we can express p as p = 3m and p = 5n, where m and n are integers.

Substituting p = 3m and p = 5n into the equation p^2 = 15q^2:
(3m)^2 = 15q^2
9m^2 = 15q^2

Simplifying further, we have:
3m^2 = 5q^2

Consequence: From the above equation, we can conclude that q^2 is divisible by 3, and therefore, q must also be divisible by 3.

Consequence: We have found that both p and q are divisible by 3, which means that p/q is not in its simplest form, contradicting our assumption that √3 * √5 is a rational number.

Conclusion: Since assuming √3 * √5 to be a rational number leads to a contradiction, we can conclude that √3 * √5 is an irrational number.

Therefore, √3 * √5 is an irrational number.