Rationalization

Preface

Learn about rationalization with examples, rationalize calculator, and how to rationalize numerator in the concept of rationalization. Check out the interactive simulations and rationalize using a calculator to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Become a champ of Rationalization in just 7 minutes!

Introduction

Rationalization is a technique that finds application in elementary mathematics. It helps in simplifying the mathematical expressions into a simpler form.

Rationalization literally means to make something more effective. Its adaption in mathematics means making the equation reduced into its more effective and simpler form.

Table of Contents

 
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Rationalization 

The word rationalize literally means making something more efficient.

Rationalization is the process of eliminating a radical or imaginary number from the denominator or numerator of an algebraic fraction.

That is, remove the radicals in a fraction so that the denominator or numerator only contains a rational number.

Radical

A radical is an expression that uses a root, such as a square root, cube root. For example, an expression of the form:

\(\sqrt{\text a+\text b}\)

is a radical.

Radicand

Radicand is the term we are finding the root of.

rationalize-example

Radical Symbol

The \(√ \) symbol means "root of". The length of the horizontal bar is important.

The length of the bar signifies that variables or constants that are a part of the root function. The variables or constants that are not under the root symbol are hence not part of the root.

Degree

The degree of a polynomial is the number of times the radicand is multiplied by itself. 2 means square root, 3 means cube root.

Further, they are referred to as 4th root, 5th root, and so on. If this is not mentioned, we take it as square root by default.

rationalize-examples, rationalize

Conjugate

A math conjugate of any binomial means another exact binomial with the opposite sign between its two terms. 

For example, conjugate of \( \text{x + y}\) is \( \text {x - y}\) or vice- versa.Thus, both these binomials are conjugates of each other.

Example

Let us have a look at the following fraction,

\[\dfrac{5}{2-\sqrt{3}}\]

The denominator in the above fraction contains a radical that needs to be rationalized.

Procedure

Suppose the denominator contains a radical expression,

\[ \text a+\sqrt{ \text b }\]

or 

\[\text a +\text i \sqrt{ \text b }\]

Here, the radical must be multiplied by its conjugate.

The procedure to rationalize an expression depends on the radical if that is a monomial or polynomial.

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How to Rationalize Numerator or Denominator?

When you look at the definition of "rationalize", it will become clearer as to what exactly rationalizing a denominator means. 

The numbers like \(\frac{1}{2}\), \(5\) and \(0.25\) are all rational numbers i.e., they can be expressed as a ratio of two integers like  \( \frac{1}{2}, \frac{5}{1}, \frac{1}{4} \) respectively . Whereas, some radicals are irrational numbers because they cannot be represented as the ratio of two integers.

Thus, the denominator needs to be rationalized to make the expression become a rational number.

The following table gives the equivalent rational values of an irrational number.

Irrational Rational
\(\dfrac{1}{\sqrt{7}}\) \(\dfrac{\sqrt{7}}{7}\)
\(\dfrac{4}{\sqrt{3}}\) \(\dfrac{4 \sqrt{3}}{3}\)
\(\dfrac{4 + \sqrt{7}}{\sqrt{7}}\) \(\dfrac{4 \sqrt{7} + 7}{7}\)

Rationalizing a Monomial Radical

Any polynomial with only one term is called a monomial. As we know, polynomials are equations or algebraic expressions which consists of variables and coefficients and has one or more terms in it.

Example: \(\sqrt{2}\), \(\sqrt{7} \text x \), \(\sqrt[3]{7 \text x }\), etc.

For a polynomial with a monomial radical in the denominator, say of the form, 

\( \text a \sqrt[ \text m ]{ \text x^ \text n }\)

\( \text n < \text m \)

The fraction must be multiplied by a quotient containing \( \text a \sqrt[ \text m ]{ \text x^{ (\text m -  \text n)} }\), both in the numerator and denominator.

This gives us the result \( \text a \sqrt[ \text m ]{ \text x ^ \text m}\), which can be replaced by \( \text x \) and hence free of the radical term.

If the denominator is linear in terms of a square root, both numerator and denominator are multiplied with that value, and the denominator product is solved.

Let us go through this technique step by step using the following example,

Example: Let us rationalize the fraction: \(\frac{2}{\sqrt{7}}\).

Step 1. Examine the fraction - The given fraction has a monomial radical \(\sqrt{7}\) in the denominator that should be rationalized. Note that the numerator can have a radical, so you don't need to worry about the numerator while examining the fraction or simplifying it.

rationalize

Step2. Multiply the fraction in the numerator and denominator by \(\sqrt{7}\).

rationalize examples

Step3. Simplify the expression as needed.

rationalize examples

Rationalizing a Binomial Radical

If the denominator has a radical expression of the form \( \text a +\sqrt{ \text b}\) or \( \text a+ \text i \sqrt{ \text b}\), the fraction must be multiplied by the conjugate of the expression i.e.,

  \( \text a -\sqrt{ \text b }\) or \( \text a - \text i \sqrt{ \text b }\) both in the numerator and denominator.

The denominator is further expanded following the suitable algebraic identities.

Example: Let us rationalize the following fraction: \[\frac{\sqrt{7}}{2 + \sqrt{7}}\]

Step1. Examine the fraction - The denominator of the above fraction has a binomial radical i.e., is the sum of two terms, one of which is an irrational number.

rationalize examples

Step2. Multiply the numerator and denominator of the fraction with the conjugate of the radical.

rationalize example

Step3. Simplify the expression as needed.

rationalize example

Rationalizing a Cube Root

This method generalizes to roots of any order.

For the given fraction with cube root radical in the denominator,

\(\dfrac{1}{\sqrt[3]{5}}\)

Step1. Examine the fraction - The fraction has a radical in the form of a cube root in the denominator.

rationalize example

Step2. Multiply the numerator and denominator of the fraction by a factor that makes the exponent of the denominator \(1\). In this case, that factor would be \(5^\frac{2}{3}\).

rationalize example

Step3. Simplify the expression as needed.

rationalize example

 
Thinking out of the box
Think Tank
  • Why do we multiply the denominator or the numerator of a fraction with a binomial radical with the conjugate of the radical and not the same term as in the case of a monomial?

Rationalization Calculator


Solved Examples

Example 1

 

 

Let us start with a simple example to rationalize:

\[\frac{1}{3 + \sqrt{10}}\]

Solution

The given fraction has a binomial radical, \( 3 + \sqrt{10}\) in the denominator that needs to be rationalized.

Multiplying both the numerator and denominator by the conjugate of the binomial radical i.e.,  \( 3 - \sqrt{10}\),

We get,

\[ \begin{align*} &= \frac{1}{3 + \sqrt{10}} \\ &= \frac{1}{3 + \sqrt{10}} \times \frac{3 - \sqrt{10}}{3 - \sqrt{10}} \end{align*}\]

\( \begin{align*}   & & \because & & ( \text a + \text b )( \text a - \text b) &=  \text a ^2 - \text b ^2 \end{align*}\)

\[ \begin{align*} &= \frac{3 - \sqrt{10}}{3^2 - \sqrt{10}^2 } \\ \\ &= \frac{3 - \sqrt{10}}{9 - 10} \\ \\ &= \frac{3 - \sqrt{10}}{-1} \end{align*} \]

 \[ \text {dividing the numerator by -1,} \]

\[  -3 + \sqrt{10} \]

\(\therefore\) \(-3 + \sqrt{10}\)
Example 2

 

 

Ira is facing some difficulties while rationalizing a fraction, \[\frac{1}{\sqrt{2} + \sqrt{5}}\] Can you help her solve it?

Solution

The given fraction, \[\frac{1}{\sqrt{2} + \sqrt{5}}\] has radical in both the terms of the denominator that need to eliminated.

Multiplying the numerator and denominator by the conjugate of the fraction,  \(\sqrt{2} - \sqrt{5}\), we get,

\[ \begin{align*} \dfrac{1}{\sqrt{2} + \sqrt{5}} \times \dfrac{\sqrt{2}-\sqrt{5}}{\sqrt{2}-\sqrt{5}} \end{align*}\]

\( \begin{align*}   & & \because & & ( \text a + \text b )( \text a - \text b) &=  \text a ^2 - \text b ^2 \end{align*}\)

\[  \begin{align*} &= \dfrac{\sqrt{2} - \sqrt{5}}{ \sqrt{2}^2 - \sqrt{5}^2} \\ \\ &= \dfrac{\sqrt{2} - \sqrt{5}}{ 2 - 5} \\ \\ &= \dfrac{\sqrt{2} - \sqrt{5}}{ -3 } \end{align*} \]

Taking \(-1 \) common out of the numerator,

\[  = -\dfrac{\sqrt{5} - \sqrt{2}}{-3 } \]

Cancelling the negative sign from both the numerator and denominator,

\[  = \dfrac{\sqrt{5} - \sqrt{2}}{ 3 } \]

\(\therefore\) \(\dfrac{\sqrt{5} - \sqrt{2}}{ 3 }\)
Example 3

 

 

Simplify the following:

\[\sqrt \frac{36}{7}\]

Solution

The fraction \(\sqrt \frac{36}{7}\) can be simplified before rationalizing as,

\[ \dfrac{6}{\sqrt7}\]

The denominator has the radical term \(\sqrt{7}\).

Thus, multiplying by \(\sqrt{7}\),

\[\dfrac{6}{\sqrt7}\times \frac{\sqrt{7}}{\sqrt{7}} \]

\[ \begin{align*}  &= \dfrac{6\sqrt7} {\sqrt{7} \times \sqrt{7}} \\ \\ &= \dfrac{6\sqrt7} {7} \end{align*}\]

\(\therefore\) \(\dfrac{6\sqrt7} {7}\)
Example 4

 

 

What will be the rationalizing factor for the fraction,

\[\frac{2 + \sqrt{7}}{2 - \sqrt{7}}\]

Solution

The rationalizing factor of a fraction is the term that needs to be multiplied with the denominator to make it a rational number.

In the given fraction,

\[\dfrac{2 + \sqrt{7}}{2 - \sqrt{7}}\]

The denominator has a binomial square root radical, \(2 - \sqrt{7}\)

Thus, the rationalization factor would be,

\[2  + \sqrt{7}\]

\(\therefore\) \(2 + \sqrt{7}\)
Example 5

 

 

The rationalization of the nth root in the denominator of a fraction can get tricky. Can you ace it in the following problem?

Rationalize:

\[\dfrac{2}{\sqrt[4]{25}}\]

Solution

The given fraction, 

\[\dfrac{2}{\sqrt[4]{25}}\]

can be written as,

\[\dfrac{2}{\sqrt[4]{5 \times 5}}\]

The denominator needs to be multiplied by:

\[\sqrt[4]{5 \times 5 }\]

to make the denominator 5.

Hence, multiplying both numerator and denominator by \(\sqrt[4]{5 \times 5 }\),

\[ \begin{align*} &= \dfrac{2}{\sqrt[4]{25}} \times \frac{ \sqrt[4]{25}}{\sqrt [4]{25}} \\ &= \dfrac{2 \times \sqrt[4]{25}}{\sqrt[4]{5 \times 5 \times 5 \times 5}} \\ &= \dfrac{2 \times \sqrt[4]{25}}{5} \\ &= \dfrac{2 \sqrt[4]{25}}{5} \end{align*} \]

\(\therefore\) \( \dfrac{2 \sqrt[4]{25}}{5}\)
 
Challenge your math skills
Challenging Questions
  • Simplify the following expression: \(\begin{align}\frac{2}{2- \sqrt{3}} + \frac{5}{3 - \sqrt{2}} - (4 \sqrt{3} + \sqrt{2})\end{align}\)

Practice Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 
 
 

Conclusion

We hope you enjoyed learning about rationalization with the simulations and practice questions. Now, you will be able to easily solve problems on rationalization with examples, rationalize calculator, and how to rationalize numerator.

You can also simplify this topic with our Math Experts in Cuemath’s LIVE and interactive online classes. Book a FREE trial class today!


Frequently Asked Questions (FAQs)

1. How do you rationalize surds?

When the denominator of a fraction contains a surd, both numerator and denominator are multiplied with that value.

The denominator product is solved, thereby giving a rationalized denominator.

2. Why do we rationalize denominators?

If the denominator is linear in terms of a radical, both numerator and denominator are multiplied with a value such that the denominator product is expanded and becomes rationalized.

For the denominator with a radical expression of the form \( \text a +\sqrt{ \text b}\) or \( \text a+ \text i \sqrt{ \text b}\), the fraction must be multiplied by the conjugate of the expression i.e.,

  \( \text a -\sqrt{ \text b }\) or \( \text a - \text i \sqrt{ \text b }\) both in the numerator and denominator.

3. How do you find the limit of a function?

The limit of a function can be calculated by rationalizing its numerator.

The numerator and denominator of the function are multiplied with the numerator's conjugate and the expression is simplified to find the limit of the function.

  
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