Tuesday, November 5, 2013

What is Radical or Shurd?




Last day you know about how to prove 6√3, Today we know what is surd or radicals
 

Positive Nth Root Of A Real Number:-

Let a be a real number and n be a positive integer. Then, a number which when raised to the power n gives a, is called the nth root of a, written as n√a or a1/n.


The symbol n called the radical sign of index n.

We denote 2√a by √a only.

Thus, √3 = 31/2 ; 3√4 = 41/3 ;  4√6 = 61/4 , etc.

Surds or Radicals:- What is surd or radical:

Let a be a rational number and n be a positive integer such that n√a is irrational, then n√a is called a surd or a radical of the order n, and a is called the radic and.

A srud of order 2 is called a quadratic surd.

A surd of order 3 is called a cubic surd.

A surd of order 4 is called a biquadratic surd.


Remarks: (i) n√a is a surd only when a is rational and n√a is irrational.

(ii) When a is irrational or n√a is rational, then n√a is not a surd.

For example:- Let us consider √3 or 31/2  

Clearly, 3 is rational, 2 is a positive integer and √3 is a quadratic surd.

Hence, √3 or 31/2 is a surd of the order 2, i.e, √3 is a a quadratic surd.

Similarly, 3√4 and 4√6 are cubic & biquadratic surd. And 5√6 are is a surd of the order 6 etc.


Take another example: √ᴨ is not a surd, since ᴨ is irrational.

3√8 is not a surd, since 3√8 =2, which is rational

Since, 3√8  = 81/3 = (23)1/3 = 2 (3 x 1/3) = 21 = 2


Remark: Every surd or radical is an irrational number, but ever irrational need not to be surd.

For e×ample: let us consider (2 + √3)2 = [22 + (√3)2 +2 × 2 ×√3]
                                                          
                                                                    = 7 +4√3

Since, 7 + 4√3 = ( 2 + √3), which is irrational.

But, 7 + 4√3 is not a surd, since 7 + 4√3 is irrational .

Hence, 7 + 4√3 is an irrational number, but it is not a surd.



Law of Radicals:- A surds can be e×pressed with fractional powers, the laws of indices are applicable to surd also. Thus, we have the following laws of radicals.
   

0
Law of Indices
Law of Radicals
1
(a1/n)n = a
(n√a)n = a
2
(ab)1/n = a1/n . b1/n
n√ab = n√a . n√b
3
(a/b)1/n = a1/n/b1/n
n√a/b = n√a/ n√b
4
(a1/n)m = (am)1/n = am/n
(n√a)m  = (n√am)
5
(a1/n)1/m = a1/mn =(a1/m)1/n
m√a n√a =  mn√a = n√a m√a


continue................

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