Question

What is the rationalizing factor of $\sqrt 2 - 1$?
A.$1 - \sqrt 2 $
B.$2 + \sqrt 2 $
C.$\sqrt 2 + 1$
D.Both A and B

Answer 1

Hint: The factor of multiplication by which rationalization is done is called the rationalizing factor. We will check if any of the mentioned options is the rationalizing factor or not. First we will first check if options upon multiplication with the given number becomes a rational number or not. If yes, then we will say it is a rationalizing factor of the given number.

Complete step by step Answer:
We are required to find the rationalizing factor of $\sqrt 2 - 1$.
For option(A), we will see if multiplication of $1 - \sqrt 2 $with $\sqrt 2 - 1$gives a rational number or not. if yes then we will say $1 - \sqrt 2 $is the rationalizing factor of the given number.
$
   \Rightarrow (\sqrt 2 - 1)(1 - \sqrt 2 ) = (\sqrt 2 - 1 - 2 + \sqrt 2 ) = (2\sqrt 2 - 3) \\
   \Rightarrow 2\sqrt 2 - 3 \\
 $
Here, we can see that the answer obtained is not a rational number.
Option(A) is incorrect as $2\sqrt 2 - 3$is not a rational number.
For option(B), again we will check if the product of both surds is a rational number or not.
$ \Rightarrow (\sqrt 2 - 1)(2 + \sqrt 2 ) = (2\sqrt 2 - 2 + 2 - \sqrt 2 ) = \sqrt 2 $
Here as well, the answer obtained is not a rational number.
Since $\sqrt 2 $is not a rational number, option(B) is also incorrect.
Now for option(C),
$(\sqrt 2 - 1)(\sqrt 2 + 1) = (2 - \sqrt 2 + \sqrt 2 - 1) = 1$
Option(C) is correct as 1 is a rational number.
Option(D) is also incorrect because option(A) and (B) are incorrect. Since, option(D) stated that both (A) and (B) are correct.

Note: If the product of two surds is a rational number, then each surd is a rationalizing factor to the other. This is exactly the same if we say that the given number is a rationalizing factor of the option(C) or vice-versa.