Question

The product of a rational number and an irrational number is
$\left( a \right)$ A natural number
$\left( b \right)$ An irrational number
$\left( c \right)$ A composite number
$\left( d \right)$ A rational number

Answer 1

Hint: In this particular question use the concept that a rational number is written in the form of $\dfrac{p}{q}$ and an irrational number cannot written in the form of $\dfrac{p}{q}$, where $q \ne 0$, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
As we know rational numbers are those which are written in the form of$\dfrac{p}{q}$, where $q \ne 0$and $\dfrac{p}{q}$is written in lowest form, such that $p$ and $q$ have not no common factors except 1.
For example: $\dfrac{2}{3}$as this fraction is written in lowest form and does not have any common factors except 1, so this is a rational number.
Now an irrational numbers are those numbers which cannot written in the form of$\dfrac{p}{q}$, where $q \ne 0$and $\dfrac{p}{q}$is written in lowest form, such that $p$and $q$have not no common factors except 1.
For example: $\sqrt 2 $ it cannot be written in the form of $\dfrac{p}{q}$, where $q \ne 0$, so this is an irrational number.
Now multiply rational and irrational number we have,
$ \Rightarrow \dfrac{2}{3} \times \sqrt 2 $
So we see that the resultant of the above multiplication is $\dfrac{{2\sqrt 2 }}{3}$, which is also a rational number.
So we can say that the product of a rational number and an irrational number is also an irrational number.
So this is the required answer.

So, the correct answer is “Option b”.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the condition of a rational and an irrational number which is all stated above so according to above properties consider any one example of both the numbers and multiply them as above we will get the resultant number is also an irrational number.