**Introduction to Rationalize the Denominator**

Before studying how to rationalize the denominator, let us understand what does rationalization means. Rationalization means to convert a given numerical expression into a rational number.

A rational number is any number which is in the form of p/q where p is not equal to 0. Its a number whose expression is as the quotient or fraction p/q of two integers, such that it has a numerator p and a non-zero denominator q.

**Why do you Rationalize Denominator?**

Now, if we look at the number line, in general, every positive and negative integer along with 0 is a rational number. But at the same time, there are various numerical expressions which cannot be justified as being a rational number. Therefore to make a numerical expression more understandable and easy for further calculation, rationalization of the number is done.

**Rationalize the Denominator: Numerical Expression**

As we discussed above, that all the positive and negative integers including zero are considered as rational numbers. To exemplify this let us take the example of number 5. 5 can be written as 5/1. Here, we can clearly see that the number easily got expressed in the form of p/q and here q is not equal to 0.

Even 0 can be written as 0/1 and we can clearly see that 0 is also a rational number as it can be expressed in the form of p/q and here q is not equal to 0.Â However, what about irrational numbers.

Lets take the example of a number such as 1/âˆš2, which is an example of an irrational number. Therefore to rationalize this numerical expression we need to follow these steps:-

**Multiply both the numerator and denominator with âˆš2**

When we multiply the with âˆš2, we get (1xâˆš2), which further equals âˆš2.

Now, multiplying the denominator with âˆš2, we get (âˆš2xâˆš2) which equals âˆš4 which equates to 2.

So now the numerical expression 1/âˆš2 becomes âˆš2/2. So now we can clearly say that âˆš2/2 is in the form of p/q and here q is not equal to 0.

Another example that we can take of an irrational number can beÂ 1/ (3-âˆš2). To convert this type of numerical expression into a rational number we need to follow the given steps:

First and foremost we need to multiply both the numerator and the denominator we the conjugate of (3-âˆš2) which equals to (3+âˆš2).

Multiplying the numerator with (3+âˆš2) we get 1 x (3+âˆš2) which equals (3+âˆš2).

Now multiplying the denominator with (3+âˆš2) we get (3-âˆš2) x (3+âˆš2) which equals

3^{2}-(âˆš2)^{2Â Â Â Â Â Â Â Â Â }

which further equates to 7.

[following (a+b)(a-b)= (a^{2}-b^{2})]

**Solved Question for You**

**Question:** Convert 1/(5+âˆš2)Â into a rational number by rationalizing the denominator.

**Answer:** Multiplying the numerator and denominator with

(5-âˆš2)Â we get:

1x (5-âˆš2) which equals (5-âˆš2) as the numerator

And

(5+âˆš2) Â x (5-âˆš2) Â which equals [(5^{2}– (âˆš2)^{2}] following (a+b)(a-b)= (a^{2}-b^{2}) which equates to (25-2) that is 23.

Therefore we get the result as 1/23 which is a rational number.

Hence, with this lesson, we learnt how to rationalize the denominator and making a numerical expression rational.

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