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We will discuss about the simple and compound surds.
Definition of Simple Surd:
A surd having a single term only is called a monomial or simple surd.
Surds which contains only a single term, are called as nominal or simple surds. For example 2β2, 2β5,2β7, 53β10, 34β12, anβx are simple surds.
More example, each of the surds β2, β7, β6, 7β3, 2βa, 5β3, mβn, 5 β 73/5 etc. is a simple surd.
Definition of Compound Surd:
The algebraic sum of two or more simple surds or the algebraic sum of a rational number and simple surds is called a compound scud.
The algebraic sum of two or more simple surds or the algebraic sum of rational numbers and simple surds are called as binominal surds or compound surds. For example 2+2β3 is a sum of one rational number 2 and one simple surd 2β3, so this is a compound surd. 2β2+2β3 is a sum of two simple surds 2β2 and 2β3, so this is also a example of compound surd. Some other examples of compound surds are 2β5β2β7, 3β10+3β12, 2βx+2βy
More example, each of the surds (β5 + β7), (β5 - β7), (5β8 - β7), (β6 + 9), (β7 + β6), (xβy - b) is a compound surd.
Note: The compound surd is also known as binomial surd. That is, the algebraic sum of two surds or a surd and a rational number is called a binomial surd.
For example, each of the surds (β5 + 2), (5 - β6), (β2 + β7) etc. is a binomial surd.
Problems on Simple Surds:
1. Arrange the following simple surds descending order.
2β3, 3β9,4β60
Solution:
The given surds are 2β3, 3β5, 4β12.
The surds are in the order of 2, 3, and 4 respectively. If we need to compare their values, we need to express them in same order. As the LCM of 2, 3, and 4 is 12, we should express the surds in order 12.
2β3 = 312 = 3612= 729112 = 12β729
3β5 = 513 = 5412= 625112 = 12β625
4β12 = 1214 = 12312 = 1728112 = 12β1728
Hence the descending order of the given surds is 4β12, 2β3, 3β5.
2. Arrange the following simple surds descending order.
22β10, 42β7, 52β3
Solution:
If we need to compare the values of the given simple surds, we have to express them in the form of pure surds. As the orders of all three surds are same we donβt need change the order.
22β10 = 2β22Γ10 = 2β4Γ10 = 2β40
42β7 = 2β42Γ7 = 2β16Γ7 = 2β112
52β3 = 2β52Γ3 = 2β25Γ3= 2β75
Hence the descending order of the given surds is 42β7, 52β3, 22β10.
Problems on Compound Surds:
1. If x = 1+2β2, then what is the value of x2β1x2?
Solution:
Given x = 1+2β2
We need find out
x2β1x2
= x2β(1x)2
As we know a2βb2=(a+b)(aβb)
We can write x2β(1x)2 as
= (x+1x)(xβ1x)
Now we will find out separately the values of x+1x and xβ1x
x+1x
= 1+2β2+11+β2
= (1+β2)2+11+β2
=1+2+2β2+11+β2
=4+2β21+β2
=2β2(1+β2)1+β2
=2β2xβ1x
=1+2β2-11+β2
=(1+β2)2β11+β2
=1+2+2β2β11+β2
=3+2β21+β2
So x2β1x2
=(x+1x)β (xβ1x)
=(2β2)(3+2β21+β2)
=6β3+81+β2
=2(3β3+4)1+β2
2. If x= β2+β3 and y = β2ββ3 then what is the value of x2βy2?
Solution:
As we know a2βb2=(a+b)(aβb)
x2βy2
= (x+y)(xβy)
Now we will find out separately the values of (x + y) and (x - y).
(x + y)
= β2+β3 + β2ββ3
= 2β2(x - y)
= β2+β3-β2ββ3
= 2β3
So x2βy2
= 2β2Γ2β3
=4β6
11 and 12 Grade Math
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