Question

a student wrote a proof about the product of two rational numbers:
Let x = a/b and let y = c/d, where a and c are defined to be integers, and b and d are nonzero integers.
By substitution, xy = ac/bd.
By applying the closure property of integers and nonzero integers on multiplication, ac is an integer and bd is a nonzero integer.
What conclusion can the student now make about the product xy?
A- the product xy is a nonzero Integer because nonzero Integers are closed on division.
B- the product xy cannot be an integer because bd is a nonzero Integer.
C- the product xy may be elther rational or Irrational because the values of a, b, c, and d are unknown.
D- the product ry is rational because it can be written as the quotient of an Integer and a nonzero Integer.​

Answer 1

pretty sure that the answer is D

Answer 2

A rational number is a number that can be expressed as a fraction of integers.

The conclusion the student can make about product xy is that:

D- the product ry is rational because it can be written as the quotient of an Integer and a nonzero Integer.​

Given that:

$x = \frac ab$

$y = \frac cd$

The product xy is:

$x \times y = \frac ab \times \frac cd$

So, we have:

$x \times y = \frac{ac}{bd}$

From the question, we understand that:

$Integers \to a, c$

This means that, ac would be an integer (by closure property)

Also:

$Non\ zero \to b,d$

This means that, bd would be non-zero i.e. $bd \ne 0$

So, we have:

$x \times y = \frac{ac}{bd}$

$x \times y = \frac{Integer}{Non\ zero}$

The quotient of an integer and a non-zero integer is always be rational.

Hence, option (d) is correct.

Read more about rational numbers at:

https://brainly.com/question/15815501