Question

which of the following correctly describes the product of a non - zero rational number and an irrational number? a. always rational b. can be either rational or irrational depending on the rationals used c. always irrational d. a combination of rational and irrational

Explanation:

By definition, a rational number can be written as $\frac{p}{q}$ where $p,q$ are integers, $q
eq0$, and an irrational number cannot be expressed this way. Using proof by contradiction: assume the product of a non-zero rational $\frac{a}{b}$ ($a,b
eq0$, integers) and irrational $x$ is rational $\frac{c}{d}$ ($c,d$ integers, $d
eq0$). Then $x=\frac{bc}{ad}$, which would make $x$ rational, contradicting $x$ being irrational. Thus the product must always be irrational.

Answer:

C. always irrational