Answer
A number is rational if it is a ratio of two integers, of which the second (the denominator) is not zero. By the closure property of the integers under subtraction, $a-b$ must be an integer, since $a$ and $b$ are both integers. By the zero product property, $b^{2}=b\times b \ne 0$, because $b\ne0$. Applying the zero product property again, $ab^{2}\ne0$, because $a\ne0$ and $b^{2}\ne0$. Finally, by the closure property of the integers under multiplication, $ab^{2}$ is an integer since $a$ and $b$ are both integers, so $\frac{a-b}{ab^{2}}$ is a ratio of two integers, the second of which is not zero. Therefore, $\frac{a-b}{ab^{2}}$ is a rational number by definition.
Work Step by Step
The three properties used in this proof are the closure of the integers under subtraction, the closure of the integers under multiplication, and the zero product property. Appendix A of the text provides a review of fundamental algebra concepts, including these.
