In this section, we will learn about pure surds as well as mixed surds.

Table of Contents

**Definition of Pure Surd**

A surd is called a pure surd if it consists of a whole rational number under the root symbol.

For example, $\sqrt{2}$ is a pure surd but $7\sqrt{2}$ is not.

Examples of Pure Surd:

(i) $\sqrt{11}, \sqrt{18}$ are pure surds.

(ii) The surd $2^{3/2}$ is an example of pure surds. Note that $2^{3/2}$ $=\sqrt{2^3}$ $=\sqrt{8}.$

**Definition of Mixed Surd**

A surd is called a mixed surd if it is a product of a rational number and a surd. More precisely, a mixed surd is a product of a rational number and a pure surd.

For example, $7\sqrt{2}$ is a mixed surd.

Examples of Mixed Surd:

(i) $2. 5^{3/2}$ is a mixed surd as it is a product of an integer $2$ and a surd $5^{3/2}$$=\sqrt{5^3}$ $=\sqrt{125}.$

(ii) If $n$ is a positive integer and $a, b$ are two rational numbers, then $a\sqrt{b},$ $a\sqrt[3]{b},$ $b\sqrt{a},$ $a\sqrt[n]{b},$ $b\sqrt[n]{a}$ are all examples of mixed surds.

**How to convert Mixed Surds into Pure surds**

Note that a mixed surd can always be transformed into a pure surd.

**Step I: **Observe there is a rational number outside the root symbol.

**Step II: **Take the rational number inside the root symbol by the following rules

\[a\sqrt[n]{b}=\sqrt[n]{a^nb}\]

For example, $5\sqrt{3}$$=\sqrt{5^2 \times 3}$ $=\sqrt{25 \times 3}$ $=\sqrt{75},$ which is a pure surd. This way one can make a mixed surd into a pure surd.

**How to convert Pure Surds into Mixed surds**

A pure surd may not always be transformed into a mixed surd. We will understand the fact with the help of examples.

**Example 1:** Convert the pure surd $\sqrt{8}$ into a mixed surd.

Note that $\sqrt{8}=\sqrt{2 \times 2 \times 2}$

$=\sqrt{2 \times 2} \times \sqrt{2}$

$=2\sqrt{2}$

So $2\sqrt{2}$ is the mixed surd-form of $\sqrt{8}.$

**Example 2:** Convert the pure surd $\sqrt[4]{8}$ into a mixed surd.

As $8$ cannot be expressed as a fourth-power of $2,$ it is not possible to simplify $\sqrt[4]{8}.$ So this surd is not a product of a rational and a surd. It’s a surd with only $8$ is under the root symbol. Thus, $\sqrt[4]{8}$ is a pure surd that cannot be converted into a mixed surd.

**Summary: **If some part of the rational number involved in a pure surd cannot be taken out after simplification, then that pure surd cannot be converted into a mixed surd. But we can always convert a mixed surd into a pure surd.

Related Topics |

**Introduction to Surds****Order of Surds****Simple & Compound Surds****Like & Unlike Surds****Surd Addition & Subtraction****Multiplication of Surds****Division of Surds****Conjugate Surds****Rationalisation of Surds**

## FAQs on Pure and Mixed Surds

**Q1: What are pure surds?**

Answer: A surd in which a whole rational number is under the root symbol, is called a pure surd. For example, √7 is a pure surd.

**Q2: Give examples of pure surds.**

Answer: √2, √5, √11 are examples of pure surds.

**Q3: Is root 3 a pure surd?**

Answer: As root 3 consists of a rational number 3 inside the root symbol, √3 is a pure surd.

**Q4: What are mixed surds?**

Answer: A surd which is a product of a rational number and an irrational number, is called a mixed surd. For example, 2√3, 4√2 are examples of mixed surds.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.