Question

The product of two irrational numbers is
A.Always irrational
B.Always rational
C.Can be both rational and irrational
D.Always integer

Answer 1

Hint: In the given question, we have been asked what is the product of two irrational numbers. For knowing the answer to this question, we need to know what irrational numbers are, not the exact textbook definition, just how they look like when they are written.

Complete step-by-step answer:
In the given question we have been asked what the product of two irrational numbers is.
Now, irrational numbers are the numbers which are non-terminating and non-repeating, i.e., there is no fixed pattern in their representation.
Some examples of irrational numbers are \[\sqrt 2 ,\sqrt[6]{{554}},\dfrac{1}{{4\sqrt {54} }},etc.\] and other non-terminating non-repeating constants like \[\pi = 3.1415....\] and \[e = 2.718...\]
Now, let us consider two irrational numbers \[\sqrt {43} \] and \[\sqrt {32} \].
Now, let us multiply them \[\sqrt {43} \times \sqrt {32} = \sqrt {1376} = \sqrt {{4^2} \times 86} = 4\sqrt {86} \], which is an irrational number.
Now, consider two other irrational numbers \[\sqrt {32} \] and \[\sqrt {18} \].
Now, let us multiply them \[\sqrt {32} \times \sqrt {18} = \sqrt {576} = 24\], which is a rational number.
Hence, the product of two irrational numbers can be either rational or irrational.
Thus, the correct option is C.

Note: The product of two irrational numbers can be rational or irrational. Whereas, the product of two rational numbers is always rational. And since the set of both whole numbers and natural numbers is a subset of rational numbers, hence, the product of two numbers of either of the sets is also always going to be a rational number.