# Surds

In Mathematics, Surds are an irrational number which cannot be represented accurately in the form of fractions or recurring decimals. So, it can be left as a square root. Surds are used in many real-time applications to make precise calculations. In this article, let us discuss the surds definition, types, six basic rules of surds, and example problems.

## Surds Definition

Surds are the square roots  (√) of numbers which don’t simplify into a whole or rational number. It cannot be accurately represented in a fraction. In other words, a surd is a root of the whole number that has an irrational value. Consider an example, √2 ≈ 1.414213, but it is more accurate to leave it as a surd √2.

## Types of Surds

The different types of surds are as follows:

• Simple Surds – A surd that has only one term is called simple surd. Example: √2, √5, …
• Pure Surds – Surds which are wholly irrational. Example: √3
• Similar Surds – The surds having the same common surds factor
• Mixed Surds – Surds that are not wholly irrational and can be expressed as a product of a rational number and an irrational number
• Compound Surds – An expression which is the addition or subtraction of two or more surds
• Binomial Surds –  A surd that is made of two other surds

## Six Rules of Surds

Rule 1:

$\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}$

Example:

To simplify √18

18 = 9 x 2 = 32 x 2, since 9 is the greatest perfect square factor of 18.

Therefore, √18 = √32 x 2

= √32 x √2

= 3 √2

Rule 2:

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Example:

√12 / 121 = √12 / √121

√22 x 3 / 11

Since 4 is the perfect square of 12

√22 x √3 / 11

= 2√3 / 11

Rule 3:

$\frac{b}{\sqrt{a}}=\frac{b}{\sqrt{a}}\times \frac{\sqrt{a}}{\sqrt{a}}=b\frac{\sqrt{a}}{a}$

You can rationalize the denominator by multiplying the numerator and denominator by the denominator.

Example:

Rationalise

5/√7

5/√7 = 5/√7 x √7/√7

Multiply numerator and denominator by √7

= 5√7/7

Rule 4:

$a\sqrt{c}\pm b\sqrt{c}=(a\pm b)\sqrt{c}$

Example:

To simplify,

5√6 + 4√6

5√6 + 4√6 = (5 + 4) √6

by the rule

= 9√6

Rule 5:

$\frac{c}{a+b\sqrt{n}}$

Multiply top and bottom by a-b √n

This rule enables us to rationalise the denominator.

Example:

To Rationalise

$\frac{3}{2+\sqrt{2}}= \frac{3}{2+\sqrt{2}}\times \frac{2-\sqrt{2}}{2-\sqrt{2}}=\frac{6-3\sqrt{2}}{4-2}$ $=\frac{6-3\sqrt{2}}{2}$

Rule 6:

$\frac{c}{a-b\sqrt{n}}$

This rule enables you to rationalise the denominator.

Multiply top and bottom by a + b√n

Example:

To Rationalise

$\frac{3}{2-\sqrt{2}}= \frac{3}{2-\sqrt{2}}\times \frac{2+\sqrt{2}}{2+\sqrt{2}}=\frac{6+3\sqrt{2}}{4-2}$ $=\frac{6+3\sqrt{2}}{2}$

### How to Solve Surds?

You need to follow some rules to solve expressions that involve surds. One method is to rationalise the denominators and it’s done by ejecting the surd in the denominator. Sometimes it may be mandatory to find the greatest perfect square factor to solve surds. This is done by considering any possible factors of the value that is square rooted. For example, you need to solve for the square root of 144. 2 x 72 gives 144 and we can have a square root of 144 without a surd. Therefore we say that 144 is the greatest perfect square factor since we cannot take the square root of a bigger number that can be multiplied by another to give 144.

### Surds Problems

Example 1:

Write down the conjugate of 5√3 + √2

Solution:

The conjugate of  5√3 + √2 is 5√3 – √2.

Example 2:

Rationalize the denominator: 1/[(8√11 )- (7√5)]

Solution:

Given:  1/[(8√11 )- (7√5)]

It is known that the conjugate of (8√11 )- (7√5) is (8√11 )+(7√5)

To rationalize the denominator of the given fraction, multiply the conjugate of denominator on both numerator and denominator.

=[1/[(8√11 )- (7√5)]]× [[(8√11 )+ (7√5)]/[(8√11 )+(7√5)]]

=[(8√11 )+ (7√5)]/[(8√11 )2-(7√5)2]

=[(8√11 )+ (7√5)]/[704- 245]

= [(8√11 )+ (7√5)]/459

Now to start practising problems and examples of surd based on above-mentioned rules, please visit BYJU’S – The Learning App.