Introduction:
When we multiply a nonzero rational number with an irrational number, the result is an irrational number. This can be explained by understanding the properties of rational and irrational numbers.

Rational Numbers:
Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. They can be written in the form of p/q, where p and q are integers and q is not equal to zero. Rational numbers include integers, fractions, and terminating or repeating decimals.

Irrational Numbers:
Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They cannot be written in the form of p/q, where p and q are integers and q is not equal to zero. Irrational numbers include square roots of non-perfect squares, cube roots of non-perfect cubes, and non-repeating decimals such as π (pi) and √2.

Multiplication of a Nonzero Rational Number and an Irrational Number:
When we multiply a nonzero rational number with an irrational number, the result will always be an irrational number. This can be understood by considering the properties of irrational numbers.

Proof:
Let's assume we have a nonzero rational number represented by the fraction p/q, where p and q are integers and q is not equal to zero. Also, let's assume we have an irrational number represented by √a, where a is a non-perfect square.

When we multiply p/q with √a, the result can be expressed as (p/q) * √a.

This can be further simplified by multiplying the numerators and denominators separately:

(p * √a) / (q)

Since a is a non-perfect square and cannot be expressed as a fraction, the result (p * √a) will also be irrational.

Therefore, the product of a nonzero rational number and an irrational number will always be an irrational number.

Conclusion:
In conclusion, when we multiply a nonzero rational number with an irrational number, the result will always be an irrational number. This is because the properties of irrational numbers do not allow them to be expressed as the ratio of two integers.