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, rationalization, and explaining why to rationalize the denominator.

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 them.

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 the 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

Why rationalize a denominator?

Answer:

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