Pure Surds and Mixed Surds

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

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.

 

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