Solution:

To prove: (√3 √5)² is an irrational number.

We know that:

- A rational number can be expressed in the form of p/q, where p and q are integers and q ≠ 0.
- An irrational number cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Let's assume that (√3 √5)² is a rational number.

Then, we can express it in the form of p/q, where p and q are integers and q ≠ 0.

(√3 √5)² = p/q

√3 √5 x √3 √5 = p/q

√15 x √15 = p/q

15 = p²/q²

p² = 15q²

This means that p² is divisible by 3 and 5.

Let's assume that p is not divisible by 3.

Then, p² will not be divisible by 3.

This contradicts the fact that p² is divisible by 3.

Similarly, let's assume that p is not divisible by 5.

Then, p² will not be divisible by 5.

This contradicts the fact that p² is divisible by 5.

Therefore, our assumption that (√3 √5)² is a rational number is wrong.

Hence, (√3 √5)² is an irrational number.

Conclusion:

- We have proved that (√3 √5)² is an irrational number.
- We have used the fact that a rational number can be expressed in the form of p/q, where p and q are integers and q ≠ 0.
- We have also used the fact that an irrational number cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
- We have assumed that (√3 √5)² is a rational number and arrived at a contradiction, which proves that (√3 √5)² is an irrational number.