MathWithNaziaa : Rationalization of Denominator

Rationalizing the denominator

Algebra is a vast area of study in mathematics, which is associated with number theory, arithmetic, geometry, and its analysis. Algebra is related to the study of symbols and their manipulation with mathematical operations. The given article is a study of the denominator and numerator, rationalisation, and explaining why to rationalise the denominator.

Why rationalize the denominator?

While performing a basic operation we rationalise a denominator to get the calculation easier and obtain a rational number as a result. In the process of rationalisation, we exclude the square roots, cube roots, or any other radical expressions from the equation. Let’s see the method of rationalisation by an example.

Numerator and Denominator

The numerator is the top part of a fraction. It explains the number of counts of part of the object present in the given fraction. The term numerator is derived from the Latin word ”enumerate” which means ‘to count’. Some example, a numerator of the fraction 2/5 explains that the given object is divided into 5 equal parts and the fraction contains two of it.

The denominator is the bottom part of a fraction. It explains how many parts of a whole object are broken into. The term denominator itself is derived from the Latin word “nomen”. The denominator indicates the type of fraction described numerator. An example of a denominator that is a denominator of a fraction is, say, 5, then that indicates that the whole object is divided into 5 equal parts.

Rationalization

Rationalization is the process of attaining a rational number as

a result of multiplying a surd with a similar surd. The other surd that multiplies is term as the rationalizing factor (RF). The whole process of rationalization is carried out by moving the square roots or cube roots from the denominator to the numerator.

For example, to rationalize an expression x + √y

Rationalizing factor x – √y

now,

= (x + √y)(x – √y) = x² – (√y)²

= x² – y

While performing a basic operation we

rationalize a denominator to get the calculation easier and obtain a rational number as a result. In the process of rationalization, we exclude the square roots, cube roots, or any other radical expressions from the equation. Let’s see the method of rationalization by an example.

Rationalize the expression (√3 – 1)/(√3 + 1)

In the expression, rationalizing factor of denominator that would be √3 – 1. Multiplying and dividing the rationalizing factor of denominator, 

 = (√3 – 1)/(√3 + 1) × (√3 – 1)/ (√3 – 1)

= (√3 – 1)²/(√3)² – 1

By the formula (a – b)² = a² – 2ab + b²

= (√3)² – 2√3 × 1 + (1)²/ (3 – 1)

= 4 – 2√3/2

Taking 2 in common in numerator

= 2(2 – √3)/2

Cancelling common factors

 = 2 – √3

Sample Problems

Question 1: Rationalize 2√3/√3

Solution:

To rationalize the expression 2√3/√3 a rationalizing factor is needed which is √3.

Now,

= 2√3/√3 × √3/√3

= 2 × 3 /3

= 2

Question 2: Rationalize (2 + √3)/√3

Solution:

To rationalize the expression, 2 + √3/√3 we need a rationalizing factor which is √3.

= 2 + √3/√3 × √3/√3

= 2√3 + 3/3

Question 3: Rationalize 1/√x

Solution:

To rationalize the expression 1/√x, rationalizing factor is required which is √x.

= 1/(√x x) √x/√x.

= √x /x

Question 4: Rationalize the expression 32/5 – √7

Solution:

32/5 – √7 to rationalise this expression, rationalizing factor is needed which is 5 + √7

= 32/5 – √7 × 5 + √7/5 + √7

= 32(5 + √7)/(5 – √7)(5 + √7)

= (160 – 32√7)/(25 + 5√7 – 5√7 – 7)

= (160 – 32√7)/32

= 5 – √7

Question 5: Rationalize the denominator 5 – √3/2 + √3

Solution:

To rationalize the expression 5 – √3/2 + √3 a rationalizing factor is needed 2 – √3.

= 5 – √3/2 + √3 × 2 – √3/ 2 – √3

= (5 – √3)(2 – √3)/ (2)² – (√3)²

= 10 – 5√3 – 2√3 + 3/4 – 3

= 13 – 7√3/1

= 13 – 7√3

Question 6: Rationalize (√3 – 1)/(√3 + 1).

Solution:

To rationalize the expression (√3 – 1)/(√3 + 1) a rationalizing factor is needed √3 – 1

= √3 – 1/√3 + 1 × √3 – 1/√3 – 1

= (√3 – 1)²/(√3)² – (1)²

= (3 + 1 – 2√3)/(3 – 1)

= (4 – 2√3)/2

= 2 – √3

Examples of Rationalization of Denominator :

Summary of rationalization of denominators

Rationalizing the denominator means eliminating the radical expressions in the denominator so that we do not have square roots, cubic roots, or any other roots. The main idea in rationalizing denominators is to multiply the original fraction by an appropriate value so that after simplifying, the denominator no longer contains radicals.

When the denominator is a monomial, we can apply the fact that:

Therefore, we can multiply both the numerator and denominator by the radical expression. After simplifying, we will obtain an expression without radicals in the denominator.

On the other hand, if the denominator is a binomial, we have to use the conjugate of the binomial. The conjugate of a binomial is equal to the same binomial, but with the sign of the middle changed.

For example, suppose we have the binomial 

 in the denominator. The conjugate of this binomial is 

The product of the binomial and its conjugate is: