The product of a nonzero rational number and an irrational number:
is undefined
cannot be determined without more information
is an irrational number
is a rational number
is an irrational number
The product of a nonzero rational number and an irrational number is an irrational number.
The product of a nonzero rational number and an irrational number is always an irrational number.
To understand why, let's first define what rational and irrational numbers are. A rational number is any number that can be expressed as a fraction (where the numerator and denominator are integers), and an irrational number is any number that cannot be expressed as a fraction and has an infinite number of non-repeating decimal places.
Now, let's consider the product of a nonzero rational number x and an irrational number y. We can express x as a fraction, say x = a/b, where a and b are integers, and b is nonzero. Since y is an irrational number, it cannot be expressed as a fraction.
When we multiply x and y, we get xy = (a/b) * y. Since y is irrational, multiplying any nonzero rational number by y will result in an irrational number. This is because multiplying any nonzero rational number by a non-zero irrational number will introduce an infinite number of non-repeating decimal places, making it impossible to express the result as a fraction.
Therefore, the product of a nonzero rational number and an irrational number is always an irrational number.