Question

\[\dfrac{2}{3}\] is a rational number whereas \[\dfrac{{\sqrt 2 }}{{\sqrt 3 }}\] is
A) Also a rational number
B) An irrational number
C) Not a number
D) A natural periodic number

Answer 1

Hint:
We will first enlist the properties of rational and irrational numbers. Then we will check whether the given number satisfies the properties of a rational number or an irrational number. We will choose the correct option accordingly.

Complete step by step solution:
We know that a rational number is a number that can be expressed in the form \[\dfrac{p}{q}\] where \[p\] and \[q\] are integers and the highest common factor of \[p\] and \[q\] is 1. Any number that is not rational or that cannot be expressed in the form \[\dfrac{p}{q}\] and the highest common factor of \[p\] and \[q\] is not 1 is called an irrational number.
 We can see that \[\dfrac{2}{3}\] is a rational number because here, \[p\] is 2 and \[q\] is 3. We know that the highest common factor of 2 and 3 is 1.
The square roots of all numbers that are not perfect squares are also irrational numbers. This is because square roots of non-perfect squares cannot be expressed in the form \[\dfrac{p}{q}\].
We will examine whether \[\dfrac{{\sqrt 2 }}{{\sqrt 3 }}\] is a rational number or not using the above properties.
We know that 2 is not a perfect square; that is, it cannot be expressed as the product of a whole number with itself. So, the square root of 2 will be an irrational number.
We also know that 3 is not a perfect square. So, the square root of 3 is also an irrational number.
We know that \[\dfrac{{\sqrt 2 }}{{\sqrt 3 }} = \sqrt {\dfrac{2}{3}} \], \[\dfrac{2}{3}\] is not a perfect square, so \[\sqrt {\dfrac{2}{3}} \] is irrational.

$\therefore $ Option B is the correct option.

Note:
We must know that it is not necessary that the division of 2 irrational numbers is also irrational. For example, \[\sqrt {50} \] and \[\sqrt 2 \] are irrational numbers. However, their division yields a rational number 5:
\[\begin{array}{l}\dfrac{{\sqrt {50} }}{{\sqrt 2 }} = \sqrt {\dfrac{{50}}{2}} \\ \Rightarrow \dfrac{{\sqrt {50} }}{{\sqrt 2 }} = \sqrt {25} \\ \Rightarrow \dfrac{{\sqrt {50} }}{{\sqrt 2 }} = 5\end{array}\]
The square root of a number (say \[a\]) when divided with the square root of another number (say \[b\]) is the same as the square root of the fraction \[\dfrac{a}{b}\] is \[\dfrac{{\sqrt a }}{{\sqrt b }} = \sqrt {\dfrac{a}{b}} \]