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Rationalization of Denominators

  • Last Updated : 18 Mar, 2021

Many times, it has been observed that we have some expressions which have radicals(i.e., an expression that uses roots, for e.g. √(x + y)) in their denominators. So, performing simple mathematical calculations like addition, subtraction, multiplication, and division are difficult on such expressions. To simplify the problem, we perform rationalization. 

As the name suggests, rationalization is a process to make a fraction rational. Rationalization is a process by which radicals in the denominator of a fraction are removed by multiplying it with an irrational number. This process makes the denominator free from radicals like square root and thus, makes the calculations easier. The number by which the denominator is multiplied to convert it into rational is called the rationalizing factor. It is important to understand that rationalization does not change the value of a number or function. It is a technique to re-write the fraction in a more acceptable form that is easy to understand. Readers may use a calculator to confirm that rationalization does not change the original value.

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Rationalizing a monomial radical



To rationalize a monomial square or cube root say a\sqrt[m]{y^n}, where n < m, we multiply the numerator and denominator by the same factor say a\sqrt[m]{y^{(m-n)}} and we get a\sqrt[m]{y^m} which can be replaced by y, so free from redical term. Or in other words, to rationalize a monomial square or cube root, we multiply the numerator and denominator by the same factor as the denominator.

Example: Let us rationalize 1/√5

So, multiple both numerator and denominator by√5

= 1/√5 × √5/√5

= √5/5

Rationalizing a Binomial Radical

If the denominator is linear and is of the form a +√b or a + i√b, rationalization comprises multiplying both the numerator and the denominator by the algebraic conjugate a – √b or a – i√b. The product is later expanded in the denominator.

Example: Let us rationalize 1/1 +√5



So, multiple both numerator and denominator by 1 – √5

\frac{1}{1+√5} \times\frac{1-√5}{1-√5}

\frac{1 - √5}{(1)^1 - (√5)^2}

\frac{1 - √5}{(1 - 5)}

\frac{√5 - 1}{4}

Sample Problem

Question 1. Interpret the meaning of 1/√3

Solution:

Since the denominator has square root in the denominator, it is a bit difficult to understand.

Let us write an equivalent expression where the denominator is a rational number.

Multiply and divide the given expression by √3.



We get,

1/√3 * √3/√3

= √3/3

It is easy to plot it on a number line.

1/√3 =√3/3 means a point which is at one third distance from 0 to √3.

So, we can interpret the meaning of 1/√3 as a point which lies at one third distance from 0 to √3.

Question 2.  Rationalize the denominator (3 +√7)/√7

Solution:

Multiply and divide the given expression by √7.

= (3 + √7)/√7 * (√7/√7)

= ((3 + √7).√7 )/√7.√7

= (3√7 + 7)/7

Question 3.  Find the value of a and b, If 1/(5 + 6√3) = a√3 + b.

Solution:

Multiply and divide the given expression by 5 – 6√3 to rationalize it.

={1/(5 + 6√3)} * {(5 – 6√3)/(5 – 6√3}

= {1 * (5 – 6√3)}/{(5 + 6√3)(5 – 6√3)}

Using the identity (a + b)(a – b) = a2 – b2

= (5 – 6√3)/{52 – (6√3)2}

=(5 – 6√3)/ 25 – 108

= (5 – 6√3)/ -83

= (6√3 – 5)/83

Given that 1/(5 + 6√3) = a√3 + b 

So, (6√3 – 5)/83 = a√3 + b

It means that a = 6/83, b = -5/83

Question 4. Given that √5 = 2.236. Find the value of 3/√5 

Solution:

Multiply and divide the given expression by √5 

=(3/√5) * (√5 /√5)

= 3 √5 /5



= 3/5 * √5

= 0.6 * 2.236

= 1.3416

Question 5.  Rationalize the denominator of 8/(√5 – √3)

Solution:

Multiply and divide the given expression by √5 + √3

= (8 * (√5 + √3))/((√5 – √3)√5 + √3))

Using the identity (a + b)(a – b) = a2 – b2

= (8√5 + 8√3)/(√52 – √32)

= 8√5 + 8√3/(5 – 3)

= 8√5 + 8√3/2

= 4√5 + 4√3 




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