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Simple and Compound Surds

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 22, 25,27, 5310, 3412, anx are simple surds.

More example, each of the surds √2, ∛7, ∜6, 7√3, 2√a, 5∛3, m∛n, 5 ∙ 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+23 is a sum of one rational number 2 and one simple surd 23, so this is a compound surd. 22+23 is a sum of two simple surds 22 and 23, so this is also a example of compound surd. Some other examples of compound surds are 2527, 310+312, 2x+2y

More example, each of the surds (√5 + √7), (√5 - √7), (5√8 - ∛7), (∜6 + 9), (∛7 + ∜6), (x∛y - 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 (√5 + 2), (5 - ∜6), (√2 + ∛7) etc. is a binomial surd.


Problems on Simple Surds:

1. Arrange the following simple surds descending order.

23, 39,460

Solution:

The given surds are 23, 35, 412.

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.

23 = 312 = 3612= 729112 = 12729

35 = 513 = 5412= 625112 = 12625

412 = 1214 = 12312 = 1728112 = 121728

Hence the descending order of the given surds is 412, 23, 35.


2. Arrange the following simple surds descending order.

2210, 427, 523

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 don’t need change the order.

2210 = 222×10 = 24×10 = 240

427 = 242×7 = 216×7 = 2112

523 = 252×3 = 225×3= 275

Hence the descending order of the given surds is 427, 523, 2210.

Problems on Compound Surds:

1. If x = 1+22, then what is the value of x21x2?

Solution:

Given x = 1+22

We need find out 

x21x2

= x2(1x)2

As we know a2b2=(a+b)(ab)

We can write x2(1x)2 as

= (x+1x)(x1x)

Now we will find out separately the values of  x+1x and  x1x

x+1x

= 1+22+11+2

= (1+2)2+11+2

=1+2+22+11+2

=4+221+2

=22(1+2)1+2

=22x1x

=1+22-11+2

=(1+2)211+2

=1+2+2211+2

=3+221+2

So x21x2

=(x+1x)(x1x)

=(22)(3+221+2)

=63+81+2

=2(33+4)1+2


2. If x= 2+3 and y = 23 then what is the value of x2y2?

Solution:

As we know a2b2=(a+b)(ab)

x2y2

= (x+y)(xy)

Now we will find out separately the values of (x + y) and (x - y).

   (x + y)

= 2+3 + 23

= 22(x - y)

= 2+3-23

= 23

So x2y2

= 22×23

=46





11 and 12 Grade Math

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