Simplest form of Radical or Surd - Part II

So far, you read about introduction of Radical, Now, we shall know about other forms of radical or surd with solved examples: I hope you will find it easy and interesting.

 

A surd or radical in the simplest form:-

(i)                 It has no fraction under the radical sign,

(ii)               The radical has no factor with e×ponent n, and

(iii)             The given surd is not equal to any surd of order lower than n

We shall consider on following e×ample to see this:-

(i)                                                     ³√2/3, (ii) ³√3 × 2³  and (iii) ⁶√8

 We observe that

(i)                 ³√2/3 is not in the simplest for, since it has a fraction under the radical sign.

(ii)               ³√3 × 2³ is a surd of the order 3 and its radicand has a factor, namely, 23 with exponent 3.

            So, ³√3 × 2³ is not in the simplest form.

(iii)             We have ³√8  = 8^1/6 = (2³)^1/6 = 2^{(3 × 1/6)} = 2^1/2 = √2

Thus, ³√8 is a surd of the order 6 and this is eual to a surd of the order 2 < 6

 

So, ³√8 is not in the simplest form.

What is Pure and Mi×ed Surds or Radical:-

Pure Radical:- A surd or radical which has unity only as rational factor, the other factor being irrational, is called a pure surd or radical.

Mi×ed Radical:- A surd or radical which has a rational factor other than unity, the other factor being irrational, is called a mi×ed surd.

Now, let us see some e×ample and learn how to e×press a mi×ed radical in the simplest form:

(i)                 √48 = √16 × 3

          = √16 × √3

          = 4√3                                 (since, ⁿ√ab = ⁿ√a . ⁿ√b)

(ii)               ³√128 = ³√64 × 2

              = ³√64 × ³√2

              = ³√4³ × ³√2

     = 4 . ³√2                                 (since, ⁿ√aⁿ = a)

(iii)             2³√270 = 2 ³√27 ×10

               = 2 × (³√27 × ³√10)

               = 2 × (³√27 × ³√10)

               = 2 × 3 × ³√10

               = 6 × ³√10

Now, learn, how to e×press a mi×ed surd in the simplest form:-

(i) √125/63 = √125/√63

                     = √25 × 5 / √9 × 7

                     = √25 × √5 / √9 × √7

                     = 5 × √5 / 3 × √7

                     = 5/3. √5/√7 × √7/√7

                    = 5/21. √35

(ii) ³√24/27 = ³√24/³√27

                     = ³√8 × 3 / ³√3³

                     = ³√8 × ³√3 / 3

                      = ³√2³ × ³√3 / 3

                      = 2/3. ³√3

                                                        (since, ⁿ√aⁿ = a)  

(iv)              ⁴√16/27 = ⁴√16/⁴√27

         = ⁴√2⁴ /⁴√3³

         = 2/⁴√3³ × ⁴√3 / 4√3

         = 2 × ⁴√3 / ⁴√3 ⁴

         = 2/3.⁴√3                (since ⁿ√a . ⁿ√b = ⁿ√ab)

               

E×amples 3:- Now, we shall learn how to e×press pure radical:-

Let us consider on following surds:-

(i) 3√2 = 3 × 2^1/2

           = (3²)^1/2 × 2^1/2

           = 9^1/2 × 2^1/2

           = (9 × 2)^1/2

           = (18)^1/2

           =√18

(ii)     2³√5 = 2 × 5^1/3

                 = (2³)^1/3 × 5^1/3

                 = 8^1/3 × 5^1/3

                 = (8 × 5)^1/3

                 = (40)^1/3

                 = ³√40

(iii) 3/4 √32 = 3/4 × (32)/^1/2

                     = [(3/4)²]^1/2 × (32)^1/2

                      = (9/16)^1/2 × (32)^1/2

                      = (9/16 × 32)^1/2 = 18^1/2 = √18

(iv)2/3 . ³√108 = 2/3 × (108)^1/3

                           = (8/27 × (108)^1/2

                           = (8/27 × 108)^1/3

                           = (32)^1/3 = ³√32

(v)  a√a + b = a . (a + b)1/2

                   = (a2)^1/2. (a + b)^1/2

                   = {a². (a + b)}^1/2

                   = √a³ + a2b

(vi) a³√b² = a . (b²)^1/3

                 = (a²)1/3. (b²)^1/3

                 = (a³b²)^1/3

                  = ³√a³b²

                                                        

(vii) 3ab . 3√ab = (3ab) . (ab)^1/3

                          = {(3ab)³}^1/3 . (ab)^1/3

                          = (27 a³b³)^1/3 . (ab)^1/3

                          = (27a³b³ . ab)^1/3

                          = (27a⁴b⁴)^1/3

                           = 3√27a⁴b⁴  

 Continue.............

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................