Question

task 2
multiplying rational and irrational numbers
in this task, youll perform multiplication on rational and irrational numbers and determine whether the sets are closed under multiplication.

part a
assume that variables x and y represent rational numbers. so, they can be written as x = \\(\frac{a}{b}\\) and y = \\(\frac{c}{d}\\), where a, b, c, and d are integers, and b and d are not equal to zero. use these variables and the closure property of integers to determine whether the product of two rational numbers is rational or irrational.

question
complete the hypothesis about the product of two rational numbers.
select the correct answer from each drop-down menu.

the product of two rational numbers is a rational number because multiplying two rational numbers is equivalent to the ratio of dropdown options: two irrational numbers, two integers, a rational number and an irrational number, which is dropdown number.

Explanation:

1. Recall the definition of a rational number: A number is rational if it can be expressed as \(\frac{m}{n}\) where \(m\) and \(n\) are integers and \(n

eq0\). Given \(x = \frac{a}{b}\) and \(y=\frac{c}{d}\) (with \(a,b,c,d\) integers, \(b,d
eq0\)), their product is \(x\times y=\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}\).

1. Analyze the numerator and denominator of the product: The numerator \(ac\) is the product of two integers (since \(a\) and \(c\) are integers, and integers are closed under multiplication, so \(ac\) is an integer). The denominator \(bd\) is also the product of two integers ( \(b\) and \(d\) are integers, so \(bd\) is an integer, and \(bd

eq0\) because \(b
eq0\) and \(d
eq0\)).

1. Determine the type of the product: Since \(\frac{ac}{bd}\) is a ratio of two integers ( \(ac\) and \(bd\)) with a non - zero denominator, it fits the definition of a rational number. So the product of two rational numbers is a rational number because multiplying two rational numbers is equivalent to the ratio of two integers, which is a rational number.

Answer:

The product of two rational numbers is a rational number because multiplying two rational numbers is equivalent to the ratio of two integers, which is a rational number.

So the correct selections are: "a rational", "two integers", "a rational".