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.
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
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.
Answer: √2, √5, √11 are examples of pure surds.
Answer: As root 3 consists of a rational number 3 inside the root symbol, √3 is a pure surd.
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.