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 a^1/n.

The symbol ⁿ√ called the radical sign of index n.

We denote ²√a by √a only.

Thus, √3 = 3^1/2 ; ³√4 = 4^1/3 ;  ⁴√6 = 6^1/4 , 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 3^1/2  

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

Hence, √3 or 3^1/2 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  = 8^1/3 = (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................