Question

Find which of the following options is correct if in each option two irrational numbers are given, and it is correct if the product of these numbers will be a rational number.
A. \[\sqrt 8 ,\sqrt 3 \]
B. \[\sqrt 6 ,\sqrt 2 \]
C. \[\sqrt 8 ,\sqrt 2 \]
D. none of these

Answer 1

Hint: Let us multiply both irrational numbers given in all options and then check whether they are rational numbers or not. A number is rational if it is a perfect number.

Complete step-by-step answer:

As we know that irrational numbers are the numbers which cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal would go on forever, without repeating.
Like some of the irrational numbers are \[\sqrt 5 {\text{, }}\sqrt 2 ,{\text{ }}\sqrt 7 {\text{, }}\sqrt {57} \].
But rational numbers are those numbers that can be expressed in the form of p/q.
Here p and q are integers, and q is not equal to 0. A rational number should have a numerator (p) and denominator (q). And rational numbers are the finite number when written in decimal.
And denominators can also be equal to 1. If denominator is 1 then that rational number is known as integers. So, integers are also rational numbers.
Some of the examples of rational numbers are \[\sqrt 4 ,{\text{ }}12,{\text{ }}\dfrac{5}{2},{\text{ }} - 15\]
So, now we had to check each option given in the question. So, that we will get the correct option in which the product of two irrational numbers is a rational number.
So, in option A.
\[\sqrt 8 \times \sqrt 3 = \sqrt {24} \] = 4.8989794…
As we know that \[\sqrt {24} \] is a never-ending number when written in decimal. So, it will not be a rational number.
Now in option B.
\[\sqrt 6 \times \sqrt 2 = \sqrt {12} \] = 3.4641016……
As we know that \[\sqrt {12} \] is a never-ending number when written in decimal. So, it will not be a rational number.
Now checking option C.
\[\sqrt 8 \times \sqrt 2 = \sqrt {16} = 4\]
As we know that \[\sqrt {16} \] is equal to 4 which is an integer (rational number). So, the product of \[\sqrt 8 ,\sqrt 2 \] will be a rational number.
Hence, the correct option will be C.

Note: Whenever we come up with this type of problem then first, we should remember that rational numbers are those numbers which have a finite number of digits when converted into decimal, but irrational numbers are those which do not have a finite number of digits when converted into a decimal number. So, we had to multiply both numbers in all of the given options and if the digits in the product are finite when converted into decimal then the product will be a rational number otherwise the product will be an irrational number. And this will be the easiest and efficient way to find the solution of the problem.