Pure and Mixed Surds

We will discuss about the pure and mixed surds.

Definition of Pure Surd:

A surd in which the whole of the rational number is under the radical sign and makes the radicand, is called pure surd.

In other words a surd having no rational factor except unity is called a pure surd or complete surd.

For example, each of the surds √7, √10, √x, ∛50, ∛x, ∜6, ∜15, ∜x, 17\(^{2/3}\), 59\(^{5/7}\), m\(^{2/13}\) is pure surd.

Definition of Mixed Surd:

A surd having a rational co-efficient other than unity is called a mixed surd.

In other words if some part of the quantity under the radical sign is taken out of it, then it makes the mixed surd.

For example, each of the surds 2√7, 3√6, a√b, 2√x, 5∛3, x∛y, 5 ∙ 7\(^{2/3}\) are mixed surd.

More examples:

√45 = \(\sqrt{3\cdot 3\cdot 5}\) = 3√5 is a mixed surd.

√32 = \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 2}\) = 2 × 2 × √2 = 4√2 is a mixed surd.

\(\sqrt[4]{162}\) = \(\sqrt[4]{ 2\cdot 3\cdot 3\cdot 3\cdot 3}\) = 3\(\sqrt[4]{2}\) is a mixed surd.


I. A mixed surd can be expressed in the form of a pure surd.

For example,

(i) 3√5 = \(\sqrt{3^{2}\cdot 5}\) = \(\sqrt{9 \cdot 5}\) = √45

(ii) 4 ∙ ∛3 = \(\sqrt[3]{4^{3}}\) ∙ ∛3 = \(\sqrt[3]{64}\) ∙ ∛3 = \(\sqrt[3]{64}\cdot 3\) = ∛192

In general, x \(\sqrt[n]{y}\) =  \(\sqrt[n]{x^{n}}\) ∙ \(\sqrt[n]{y}\) = \(\sqrt[n]{x^{n}y}\)

II. Sometimes a given pure surd can be expressed in the form of a mixed  surd.

For example,

(i) √375 = \(\sqrt{5^{3}\cdot 3}\) = 5√15;

(ii) ∛81 = \(\sqrt[3]{3^{4}}\) = 3∛3

(iii) ∜64 = \(\sqrt[4]{2^{6}}\) = 2\(\sqrt[4]{2^{2}}\)= 2\(\sqrt[4]{4}\)

But ∛20 can't be expressed in the form of mixed surd.

11 and 12 Grade Math

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