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 \underline{\quad} number because multiplying two rational numbers is
equivalent to the ratio of \underline{\quad}, which is \underline{\quad} number.

Explanation:

1. Recall the definition of a rational number: A rational number can be expressed as the ratio of two integers (with the denominator non - zero). Given \(x=\frac{a}{b}\) and \(y = \frac{c}{d}\) where \(a,b,c,d\) are integers and \(b

eq0,d
eq0\).

1. Calculate the product \(x\times y\): \(x\times y=\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}\).
2. Analyze the numerator and denominator of the product: Since \(a\) and \(c\) are integers, their product \(ac\) is also an integer (by the closure property of integers under multiplication). Similarly, \(b\) and \(d\) are non - zero integers, so their product \(bd\) is a non - zero integer (because the product of two non - zero integers is non - zero).
3. Determine the type of the product: The number \(\frac{ac}{bd}\) is a ratio of two integers (with the denominator non - zero), so it is a rational number.

For the first drop - down: The product of two rational numbers is a "rational" number.

For the second drop - down: Multiplying two rational numbers is equivalent to the ratio of "two integers (the product of the numerators and the product of the denominators)" (i.e., \(ac\) and \(bd\) which are integers).

For the third drop - down: Since \(\frac{ac}{bd}\) is a ratio of two integers with a non - zero denominator, it is a "rational" number.

Answer:

The product of two rational numbers is \(\boldsymbol{\text{rational}}\) number because multiplying two rational numbers is equivalent to the ratio of \(\boldsymbol{\text{two integers (the product of the numerators and the product of the denominators)}}\), which is \(\boldsymbol{\text{rational}}\) number.