How to Rationalize the Denominator?

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

3²-(√2)²                 

which further equates to 7.

[following (a+b)(a-b)= (a²-b²)]

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 [(5²– (√2)²] following (a+b)(a-b)= (a²-b²) 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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