• define and recognize surds as irrational numbers;
  • rationalize;
  • perform basic operations on surds
  • solve problems


Surds are numbers written in the root form i.e. \sqrt{x} to express its exact value. They are irrational numbers and are mostly expressed as root of rational numbers. The intent of solving surds is to make irrational numbers rational.

Rational and Irrational Numbers

Rational Numbers

In mathematics, we deal more with rational numbers. Rational numbers are all numbers that can be expressed in the form \frac{a}{b}, where a and b are integers like 3 and 6 and b\neq 0.

So \dpi{150} 0.5{\color{Green} } is rational since it can be in the form \dpi{150} \frac{1}{2{\color{DarkGreen} }}

The value \dpi{150} 5{\color{Green} } is rational, can be written in the form \dpi{150} \frac{5}{1{\color{Green} }}

Rational numbers also includes recurring or repeated decimals. Example \dpi{150} \frac{13}{3}=4.33333.....{\color{Green} } The dot shows that the digit 3 repeats indefinitely. Also \dpi{150} \frac{7}{4}=1.75{\color{Green} } is rational.

Irrational Numbers

Irrational simply means “not rational”. They are numbers that are not rational. And so, cannot  be written as a fraction. Irrational numbers when simplified, goes on indefinitely. In other words, we can say, they continue indefinitely without recurring or repeating. Examples are roots like

\dpi{150} \sqrt{3}=1.7320508075688{\color{Green} }\dpi{150} \sqrt{2}=1.4142135623730{\color{Green} } and numbers like \dpi{150} \sqrt{27{\color{Green} }} which is irrational, but can be made rational using surds. We should note that not all roots are irrational, roots like \dpi{150} \sqrt{4}=2 and \dpi{150} \sqrt{9}=3  etc. are all rational numbers.

Recall we said earlier on that the main intent of solving surds is to make irrational numbers rational. Having learnt the differences between this two types of numbers, let’s take a further step in finding out the rules that should guide us as we solve surds.

Surds Rules

Rule 1: \dpi{150} \sqrt{x\times y}=\sqrt{x} \times \sqrt{y}

Example: Simplify \dpi{150} \sqrt{54} . Note that \dpi{150} \sqrt{54{\color{Green} }} is an irrational number. So we make it rational.

\dpi{150} \sqrt{54} =\sqrt{9\times 6{\color{Green} }}

So \dpi{150} \sqrt{9}=3, so the solution becomes \dpi{150} \sqrt[3]{6}

Rule 2: \dpi{150} \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}

Note: \dpi{150} \sqrt{a}-\sqrt{b}\neq \sqrt{a-b} and \dpi{150} \sqrt{a} +\sqrt{b}\neq \sqrt{a+b} . The example below will show how to solve a question using this rule.

Example: Simplify \dpi{150} \frac{\sqrt{25}}{\sqrt{16 }}

\dpi{150} \frac{\sqrt{25}}{\sqrt{16}}=\sqrt{\frac{25}{16}}=\frac{5}{4}

Adding and Subtracting Similar Surds

Surds in the same basic form can be added and subtracted. The addition and subtraction can only be possible if they have same roots. So mixed surds such as \dpi{150} \sqrt[3]{2}+\sqrt[2]{3} and \dpi{150} \sqrt[5]{3}-\sqrt[4]{5} cannot be simplified further. See the example below.

Example: Simplify \dpi{150} \sqrt{28}+\sqrt{63}

To solve the above question, we have to reduce all the surds to their basic forms, i.e. \dpi{150} \sqrt{28}=\sqrt{4\times 7} and \dpi{150} \sqrt{63}=\sqrt{9\times 7}

So \dpi{150} \sqrt{28}+\sqrt{63}=\sqrt{4\times 7}+\sqrt{9\times 7}

This will give \dpi{150} \sqrt[2]{7}+\sqrt[3]{7}. Since they have same roots, they can be added, so \dpi{150} \sqrt[2]{7}+\sqrt[3]{7}=\sqrt[5]{7}

Rationalising The Denominator of Surds

A surd such as \dpi{150} \frac{\sqrt{3}}{2}, cannot be simplified, rather \dpi{150} \frac{2}{\sqrt{3}} can be written in a more convenient form. In the present form, it is irrational. To make it rational, we have to multiply the numerator and denominator by \dpi{150} \sqrt{3}

This will give us \dpi{150} \frac{2}{\sqrt{3}}=\frac{2}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt[2]{3}}{3}

Note: \dpi{150} \sqrt{3}\times \sqrt{3}=3

This removes the irrational number \dpi{150} \sqrt{3} from the denominator. This process is called rationalising the denominator or simply rationalization of surd. 

Conjugate of Surds

In surds, the rationalisation of the denominator is meant to make the denominator rational. This is done by multiplying both the numerator and the denominator by the “CONJUGATE SURD“.  

Two surds whose products results in a rational number are called conjugates. For example, if \dpi{150} (\sqrt{x}+\sqrt{y}) is a surd, the conjugate will be \dpi{150} (\sqrt{x}-\sqrt{y}). This is so since the product \dpi{150} (\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})=x-y.

For example, the conjugate of \dpi{150} (\sqrt[-3]{2}+\sqrt{7}) is \dpi{150} (\sqrt[-3]{2}-\sqrt{7}). This is because the product (multiplication) of both results to a rational number which is \dpi{150} 11.

\dpi{150} (\sqrt[-3]{2}+\sqrt{7})(\sqrt[-3]{7}-\sqrt{7})=(\sqrt[-3]{2})^{2}-(\sqrt{7})^{2}=18-7=11

Expression such as \dpi{150} \frac{1}{a-\sqrt[b]{c}} , \dpi{150} \frac{1}{\sqrt[a]{b}+c} etc. can be solved using the conjugate of the surds to rationalising the denominator.

Example: Simplify \dpi{150} \frac{1}{(1-\sqrt{3})^{2}}

To solve this question, we will first simplify the denominator, since it is squared.

\dpi{150} (1-\sqrt{3})^{2}=(1-\sqrt{3})(1-\sqrt{3})

\dpi{150} 1^{2}-\sqrt{3}-\sqrt{3}+3=1-\sqrt[2]{3}+3=4-\sqrt[2]{3}. With this the expression will become \dpi{150} \frac{1}{4-\sqrt[2]{3}}

The conjugate of \dpi{150} 4-\sqrt[2]{3} is \dpi{150} 4+\sqrt[2]{3}.

Multiply the numerator and the denominator by the conjugate.

\dpi{150} \frac{1}{4-\sqrt[2]{3}}\times \frac{4+\sqrt[2]{3}}{4+\sqrt[2]{3}}=\frac{4+\sqrt[2]{3}}{(4-\sqrt[2]{3})(4+\sqrt[2]{3})}

\dpi{150} \frac{4+\sqrt[2]{3}}{16+\sqrt[8]{3}-\sqrt[8]{3}-4\times 3}=\frac{4+\sqrt[2]{3}}{16-12}

\dpi{150} \frac{4+\sqrt[2]{3}}{4}=1+\frac{\sqrt{3}}{2}

Conclusively, from this lesson, we have been able to learn the following: 

  1. the definition and rules we must work with when solving questions on surds accurately;
  2. how to add and subtract simple surds questions, which if applied can help you solve tougher questions;
  3. learn how to rationalise the denominator of surds;
  4. know what conjugate are and how to carry out the rationalisation of questions involving conjugates.

Click to learn about Permutation and Combination. In addition, download a Textbook with more explanation on more topics in mathematics with JAMB, WAEC (SSCE, GCE) past questions and answers.


Solving surds questions can be very easy and interesting as long as you familarize youself with the rules guiding its solutions. The summary of all that have been discussed in the article are:

  1. Irrational numbers of the form \dpi{150} \sqrt[n]{a} whare \dpi{150} a is a non perfect square and \dpi{150} n is a positive integer greater tha 1 are called SURDS;
  2. To rationalize a surd means to make the denominator rational;
  3. \dpi{150} \sqrt{a} \times \sqrt{b}=\sqrt{ab} ;
  4. \dpi{150} \frac{\sqrt{a}}{\sqrt{b}} =\sqrt{\frac{a}{b}}  and 
  5. And above all we have seen that the congugate of a surd like \dpi{150} 1-\sqrt[2]{3} = 1 +\sqrt[2]{3}

Video Lessons

Also, download Worksheets on:

Thank you for reading this mathematics article and hope it was educative. if you desire to write to Us or comment on this article, we do appreciate hearing from you! Click HERE to join our discussion forum. In addition, click the various links below to follow and subscribe to our various social media handles.

Join our over 20,000+ readers to receive articles on mathematics. To get information on Website design, Digital Marketing, External Examinations in Nigeria, and lots more? Visit our blog page to read articles on these and lots more.

July 13, 2021

New Track

New Track is a leading brand in educational consulting. We train Students preparing to take external examinations with academic ebooks, YouTube videos, discussion forums, etc.
We publish posts on topics ranging from the latest News for JAMB, WEAC, NECO, etc, website design, digital marketing, tropical current affairs mathematics etc. We at New Track are poised to give you the best of the best.
Look forward to seeing more students join this Community!

Leave a Reply

Your email address will not be published.