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We will discuss about the pure and mixed surds.
If x is a positive integer with nth root, then nβx is a surd of nth order when the value of nβx is irrational. In nβx 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 β7, β10, βx, β50, βx, β6, β15, βx, 172/3, 595/7, m2/13 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 2β2, 2β5,2β7, 2β12, 3β15, 5β30, 7β50, nβx 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 2β7, 3β6, aβb, 2βx, 5β3, xβy, 5 β 72/3 are mixed surd.
More examples:
β45 = β3β
3β
5 = 3β5 is a mixed surd.
β32 = β2β
2β
2β
2β
2 = 2 Γ 2 Γ β2 = 4β2 is a mixed surd.
4β162 = 4β2β
3β
3β
3β
3 = 34β2 is a mixed surd.
But surds can have rational co-efficient other than unity. Like 2β2, 53β10, 34β12, anβx 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 2β12 is pure surd which can be written as 42β3 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 2β7, 3β11, 53β10, 34β15 these are mixed surds, we will see now how it can be converted into pure surds.
2β7= 2β22Γ7= 2β4Γ7= 2β28β¦..Pure Surd.
3β11= 2β32Γ11= 2β9Γ11= 2β99β¦..Pure Surd.
53β10= 3β53Γ10= 3β125Γ10 = 3β1250..Pure Surd.
34β15= 4β34Γ15= 4β81Γ15 = 4β1215β¦Pure Surd.
More example,
(i) 3β5 = β32β 5 = β9β 5 = β45
(ii) 4 β β3 = 3β43 β β3 = 3β64 β β3 = 3β64β
3 = β192
In general, x nβy = nβxn β nβy = nβxny
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.
2β12= 2β4Γ3= 2β22Γ3= 22β3β¦.Mixed Surd.
2β50= 2β25Γ2= 2β52Γ2= 52β2β¦.Mixed Surd.
3β81= 3β27Γ3= 3β33Γ3= 33β3β¦.Mixed Surd.
4β1280= 4β256Γ5= 4β44Γ5= 44β5β¦.Mixed Surd.
More example,
(i) β375 = β53β 3 = 5β15;
(ii) β81 = 3β34 = 3β3
(iii) β64 = 4β26 = 24β22= 24β4
But β20 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 canβt be converted into mixed surds.
Like 2β15, 3β30, 2β21, 4β40 are the examples of pure surds which canβt 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 canβt be expressed in the form of mixed surds.
In general the way of expressing a mixed surd to a pure surd is given below.
anβx= nβanΓx.
Solved example on Pure and Mixed Surds:
Express the following surds in the form of pure surds.
3β7, 23β5, 54β10
Solution:
3β7= 2β32Γ7= 2β9Γ7= 2β63β¦..Pure Surd.
23β5= 3β23Γ5= 3β8Γ5 = 3β40..Pure Surd.
54β10= 4β54Γ10= 4β625Γ10 = 4β6250β¦Pure Surd.
β Surds
11 and 12 Grade Math
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