Question

What is meant by rationalization of a surd? Write the rationalizing factor $a\sqrt{x+y}$.

Answer 1

Hint: Try to relate the definition of rationalization of surds with the procedure of converting any irrational term in any expression to rational term. Now, use the above definition to get the rationalizing factor of $a\sqrt{x+y}$.

Complete step-by-step answer:
A surd is especially the irrational root of an integer. Surds are irrational numbers but if you multiply a surd with a suitable factor, the result of the multiplication will be a rational number. This is the basic principle involved in the rationalization of surds.
The factor of multiplication by which rationalization is done, is called the rationalizing factor. Hence, if the product of two surds is a rational number, then each surd is the rationalizing factor to each other. The procedure of multiplying a surd by another surd to get a rational number is called rationalisation. The operands involved in rationalisation are called rationalizing factor.
Like if $\sqrt{2}$ is multiplied by $\sqrt{2}$, it will become 2, which is a rational number. Hence $\sqrt{2}$ is the rationalizing factor of $\sqrt{2}$.
In other words, the process of reducing a given surd to a rational form after multiplying it by a suitable surd is known as rationalization.
Another example of rationalization would be given as:

Eg: $\left( \sqrt{2}+1 \right)$ is the rationalizing factor of $\left( \sqrt{2}-1 \right)$ to convert the irrational form of number to rational in the denominator.In this type we have to take conjugate of surd means we have to take opposite sign of the surd to get rational number.If we won't take conjugate of surd then resulting multiplication will be irrational number.

Hence, the rationalizing factor of $a\sqrt{x+y}$ can be given as $\sqrt{x+y}$, since, multiplying both the terms we will get a (x + y) which is a rational number.

Note: One may use the rationalization property with the question of type
$\dfrac{1}{\sqrt{x-3}},\dfrac{5+x}{\sqrt{{{x}^{2}}-3}}$
Where we need to remove the root factor from the denominator or may be from the numerator as well with some questions.
Rationalizing terms will be the same to the term which we are going to rationalize, as the square of any number in root will make it a rational number.For some types of surds we have to take conjugate of surd means we have to take opposite sign of the surd to get rational number For Ex. $\left( \sqrt{5}-6 \right)$ rationalizing factor for this surd is conjugate of that surd i.e$\left( \sqrt{5}+6 \right)$,if we multiply both we get rational number.
Rationalizing property is used with a lot of questions in limit and continuity chapter.
Example:
$\underset{x\to 0}{\mathop{\lim }}\,\dfrac{{{x}^{2}}-9}{\sqrt{x-3}}$ .