Subscribe to our YouTube channel for the latest videos, updates, and tips.


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.


Note:

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. 

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

Solution:

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

 Surds







11 and 12 Grade Math

From Pure and Mixed Surds to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Terms Related to Simple Interest | Simple Interest Formula | Principal

    Jun 18, 25 02:57 PM

    In terms related to simple interest we will learn all the terms related to simple interest. The terms related to simple interest are Interest, Principal, Amount, Simple Interest, Time or period of tim…

    Read More

  2. Introduction to Simple Interest | Definition | Formula | Examples

    Jun 18, 25 01:50 AM

    Simple Interest
    In simple interest we will learn and identify about the terms like Principal, Time, Rate, Amount, etc. PRINCIPAL (P): The money you deposit or put in the bank is called the PRINCIPAL.

    Read More

  3. 5th Grade Profit and Loss Percentage Worksheet | Profit and Loss | Ans

    Jun 18, 25 01:33 AM

    5th Grade Profit and Loss Percentage Worksheet
    In 5th grade profit and loss percentage worksheet you will get different types of problems on finding the profit or loss percentage when cost price and selling price are given, finding the selling pri…

    Read More

  4. Worksheet on Profit and Loss | Word Problem on Profit and Loss | Math

    Jun 18, 25 01:29 AM

    Worksheet on Profit and Loss
    In worksheet on profit and loss, we can see below there are some different types of questions which we can practice in our homework.

    Read More

  5. Calculating Profit Percent and Loss Percent | Profit and Loss Formulas

    Jun 15, 25 04:06 PM

    In calculating profit percent and loss percent we will learn about the basic concepts of profit and loss. We will recall facts and formula while calculating profit percent and loss percent. Now we wil

    Read More