Question

If the product of two irrational numbers is rational, then which of the following can be concluded ? 

(A) The ratio of the greater and the smaller numbers is an integer. 

(B) The sum of the numbers must be rational. 

(C) The excess of the greater irrational number over the smaller irrational number must be rational. 

(D) None of these

Answer 1

Correct option is (D) None of these

(A) Let both rational numbers are \((\sqrt7+\sqrt3)\;and\;(\sqrt7-\sqrt3).\)

\(\therefore\) \((\sqrt7+\sqrt3)(\sqrt7-\sqrt3)\) = 7 - 3 = 4 is a rational number.

But \(\frac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}=\frac{(\sqrt7+\sqrt3)(\sqrt7+\sqrt3)}{(\sqrt7-\sqrt3)(\sqrt7+\sqrt3)}\) \(=\frac{7+3+2\sqrt{21}}{7-3}=\frac{5+\sqrt{21}}{2}\) which is an irrational number.

Thus, the ratio of greater and smaller numbers may not be an integer while the product of two irrational numbers is a rational number.

(B) For given center example.

Sum \(=(\sqrt7+\sqrt3)+(\sqrt7-\sqrt3)=2\sqrt7\) which is an irrational number.

Thus, the sum of both irrational numbers may not be a rational number while the product of two irrational numbers is a rational number.

(C) Difference greater irrational number and smaller irrational number \(=(\sqrt7+\sqrt3)-(\sqrt7-\sqrt3)=2\sqrt3\) which is an irrational number.

Here, excess of greater irrational number over the smaller irrational number is \(2\sqrt3\) which is an irrational number.

Thus, excess of the greater irrational number over the smaller irrational number may be irrational while the product of two irrational numbers is a rational number.

Hence, if the product of two irrational numbers is a rational.

Then none of the above can be concluded for sure, it may be happen or may be not.

Answer 2

(D) None of these