Introduction:
√2 and √5 are both irrational numbers. The product of two irrational numbers is also an irrational number. In this case, √2 * √5 is irrational. In this response, we will prove that √2 * √5 is irrational.

Proof:
Assume that √2 * √5 is a rational number. This means that we can write it in the form of p/q where p and q are integers and q ≠ 0.

√2 * √5 = p/q

Squaring both sides, we get:

2 * 5 = p^2/q^2

10q^2 = p^2

This means that p^2 is a multiple of 10, which implies that p is also a multiple of 10.

Let p = 10k, where k is an integer.

Substituting this value of p in the equation 10q^2 = p^2, we get:

10q^2 = (10k)^2

10q^2 = 100k^2

q^2 = 10k^2

This means that q^2 is a multiple of 10, which implies that q is also a multiple of 10.

But we assumed that p and q have no common factors. This is a contradiction since both p and q are multiples of 10.

Therefore, our assumption that √2 * √5 is a rational number is false.

Conclusion:
Hence, we have proved that √2 * √5 is an irrational number.