**SURDS LESSON OBJECTIVES**

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

### Definition

Surds are numbers written in the root form i.e. 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 , where a and b are integers like 3 and 6 and .

So is rational since it can be in the form

The value is rational, can be written in the form

Rational numbers also includes recurring or repeated decimals. Example The dot shows that the digit 3 repeats indefinitely. Also 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

, and numbers like which is irrational, but can be made rational using surds. We should note that not all roots are irrational, roots like and 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: .

**Example**: Simplify . Note that is an irrational number. So we make it rational.

So , so the solution becomes

Rule 2:

Note: and . The example below will show how to solve a question using this rule.

**Example**: Simplify

### 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 and cannot be simplified further. See the example below.

**Example**: Simplify

To solve the above question, we have to reduce all the surds to their basic forms, i.e. and

So

This will give . Since they have same roots, they can be added, so

### Rationalising The Denominator of Surds

A surd such as , cannot be simplified, rather 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 .

This will give us

Note:

This removes the irrational number 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 is a surd, the conjugate will be . This is so since the product .

For example, the conjugate of is . This is because the product (multiplication) of both results to a rational number which is .

Expression such as , etc. can be solved using the conjugate of the surds to rationalising the denominator.

**Example**: Simplify

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

. With this the expression will become

The conjugate of is .

Multiply the numerator and the denominator by the conjugate.

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

- the definition and rules we must work with when solving questions on surds accurately;
- how to add and subtract simple surds questions, which if applied can help you solve tougher questions;
- learn how to rationalise the denominator of surds;
- 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.**

### Summary

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:

- Irrational numbers of the form whare is a non perfect square and is a positive integer greater tha 1 are called SURDS;
- To rationalize a surd means to make the denominator rational;
- ;
- and
- And above all we have seen that the congugate of a surd like

### Video Lessons

Also, download Worksheets on:

**Addition of Numbers for Grade 4****Subtraction of Numbers for Grade 4****Prime Numbers for Grade 5****Whole Number (Counting, writing and reading) for Garde 6 or 7 or JSS 1**

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