Pure and Mixed Surds

We will discuss about the pure and mixed surds.

If x is a positive integer with nth root, then \(\sqrt[n]{x}\) is a surd of nth order when the value of \(\sqrt[n]{x}\) is irrational. In \(\sqrt[n]{x}\) expression n is the order of surd and x is called as radicand.

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. 

If a surd has the whole number under the radical or root sign and the whole rational number makes a radicand, is called as pure surd. Pure surd has no rational factor except unity. For example \(\sqrt[2]{2}\), \(\sqrt[2]{5}\),\(\sqrt[2]{7}\), \(\sqrt[2]{12}\), \(\sqrt[3]{15}\), \(\sqrt[5]{30}\), \(\sqrt[7]{50}\), \(\sqrt[n]{x}\) all are pure surds as these have rational numbers only under radical sign or the whole expression purely belongs to a 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.

But surds can have rational co-efficient other than unity. Like \(2\sqrt{2}\), \(5\sqrt[3]{10}\), \(3\sqrt[4]{12}\), \(a\sqrt[n]{x}\) are surds where with pure surds some rational numbers is there in the form of rational co-efficient which are 2,5,3,a respectively. This type of surds where the rational co-efficients are not unity is called as mixed surds. From pure surds if some numbers can be taken out of radical sign, then it becomes mixed surds. Like \(\sqrt[2]{12}\) is pure surd which can be written as \(4\sqrt[2]{3}\) and this becomes a mixed surd.


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

Mixed surds can be expressed in the form of pure surds. Because if we make rational co-efficient under radical sign, it will become a pure surd. For example \(2\sqrt{7}\), \(3\sqrt{11}\), \(5\sqrt[3]{10}\), \(3\sqrt[4]{15}\) these are mixed surds, we will see now how it can be converted into pure surds.

\(2\sqrt{7}\)= \(\sqrt[2]{2^{2}\times 7}\)= \(\sqrt[2]{4\times 7}\)= \(\sqrt[2]{28}\)…..Pure Surd.

\(3\sqrt{11}\)= \(\sqrt[2]{3^{2}\times 11}\)= \(\sqrt[2]{9\times 11}\)= \(\sqrt[2]{99}\)…..Pure Surd.

\(5\sqrt[3]{10}\)= \(\sqrt[3]{5^{3}\times 10}\)= \(\sqrt[3]{125\times 10}\) = \(\sqrt[3]{1250}\)..Pure Surd.

\(3\sqrt[4]{15}\)= \(\sqrt[4]{3^{4}\times 15}\)= \(\sqrt[4]{81\times 15}\) = \(\sqrt[4]{1215}\)…Pure Surd.

More 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.

Pure surds may be expressed in the form of mixed surds also, if some value under radical sign can be taken out as rational co-efficient. In the following examples we will see how a pure surd can expressed in the form of mixed surd.

\(\sqrt[2]{12}\)= \(\sqrt[2]{4\times 3}\)= \(\sqrt[2]{2^{2}\times 3}\)= \(2\sqrt[2]{3}\)….Mixed Surd.

\(\sqrt[2]{50}\)= \(\sqrt[2]{25\times 2}\)= \(\sqrt[2]{5^{2}\times 2}\)= \(5\sqrt[2]{2}\)….Mixed Surd.

\(\sqrt[3]{81}\)= \(\sqrt[3]{27\times 3}\)= \(\sqrt[3]{3^{3}\times 3}\)= \(3\sqrt[3]{3}\)….Mixed Surd.

\(\sqrt[4]{1280}\)= \(\sqrt[4]{256\times 5}\)= \(\sqrt[4]{4^{4}\times 5}\)= \(4\sqrt[4]{5}\)….Mixed Surd.

More 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.

But when there is no multiplication factor under the radical sign which can be taken out, that surds can’t be converted into mixed surds.

Like \(\sqrt[2]{15}\), \(\sqrt[3]{30}\), \(\sqrt[2]{21}\), \(\sqrt[4]{40}\) are the examples of pure surds which can’t be expressed in the form of mixed surds.

So all mixed surds can be expressed in the form of pure surds but all pure surds can’t be expressed in the form of mixed surds.

In general the way of expressing a mixed surd to a pure surd is given below.

\(a\sqrt[n]{x}\)= \(\sqrt[n]{a^{n}\times x}\).

Solved example on Pure and Mixed Surds:

Express the following surds in the form of pure surds. 

\(3\sqrt{7}\), \(2\sqrt[3]{5}\), \(5\sqrt[4]{10}\)


\(3\sqrt{7}\)= \(\sqrt[2]{3^{2}\times 7}\)= \(\sqrt[2]{9\times 7}\)= \(\sqrt[2]{63}\)…..Pure Surd.

\(2\sqrt[3]{5}\)= \(\sqrt[3]{2^{3}\times 5}\)= \(\sqrt[3]{8\times 5}\) = \(\sqrt[3]{40}\)..Pure Surd.

\(5\sqrt[4]{10}\)= \(\sqrt[4]{5^{4}\times 10}\)= \(\sqrt[4]{625\times 10}\) = \(\sqrt[4]{6250}\)…Pure Surd.


11 and 12 Grade Math

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