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Let
a be a non-zero rational number and
b be an irrational number. Is
(a)/(b) rational or irrational?
Choose 1 answer:
(A) Rational
(B) Irrational
(C) It can be either rational or irrational

Let $a$ be a non-zero rational number and $b$ be an irrational number. Is $\frac{a}{b}$ rational or irrational? Choose $1$ answer: $\newline$ (A) Rational $\newline$ (B) Irrational $\newline$ (C) It can be either rational or irrational

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Properties of operations on rational and irrational numbers

Full solution

Q. Let

a

be a non-zero rational number and

b

be an irrational number. Is

\frac{a}{b}

rational or irrational? Choose

1

answer:

\newline

(A) Rational

\newline

(B) Irrational

\newline

(C) It can be either rational or irrational

Identify nature of numbers: Identify the nature of the numbers involved. $\newline$ $a$ is a non-zero rational number, which means it can be expressed as the ratio of two integers, where the denominator is not zero. $\newline$ $b$ is an irrational number, which means it cannot be expressed as the ratio of two integers.
Division of rational by irrational: Consider the division of a rational number by an irrational number. $\newline$ The division of a rational number by an irrational number results in an irrational number. This is because if $(a)/(b)$ were rational, then we could express $(a)/(b)$ as the ratio of two integers, say $p/q$ . Multiplying both sides by $b$ would give $a = (p/q) \cdot b$ . Since $a$ and $q$ are both rational, $(p/q) \cdot b$ would have to be rational for $a$ to be rational. However, this would imply that $b$ is a factor of a rational number, which would make $b$ rational. This contradicts the fact that $b$ is irrational.
Conclude nature of $(a)/(b)$ : Conclude the nature of $(a)/(b)$ . $\newline$ Since the division of a non-zero rational number by an irrational number results in an irrational number, $(a)/(b)$ is irrational.

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