**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. However, what about irrational numbers.

Lets take the example of a number such as 1/√2, 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 √2**

When we multiply the with √2, we get (1x√2), which further equals √2.

Now, multiplying the denominator with √2, we get (√2x√2) which equals √4 which equates to 2.

So now the numerical expression 1/√2 becomes √2/2. So now we can clearly say that √2/2 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 be 1/ (3-√2). 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-√2) which equals to (3+√2).

Multiplying the numerator with (3+√2) we get 1 x (3+√2) which equals (3+√2).

Now multiplying the denominator with (3+√2) we get (3-√2) x (3+√2) which equals

3^{2}-(√2)^{2 }

which further equates to 7.

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

**Solved Question for You**

**Question:** Convert 1/(5+√2) into a rational number by rationalizing the denominator.

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

(5-√2) we get:

1x (5-√2) which equals (5-√2) as the numerator

And

(5+√2) x (5-√2) which equals [(5^{2}– (√2)^{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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