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Order of a Surd

The order of a surd indicates the index of root to be extracted.

In na, n is called the order of the surd and a is called the radicand.

For example: The order of the surd 5z is 5.

(i) A surd with index of root 2 is called a second order surd or quadratic surd.

The surds which have the indices of root 2 are called as second order surds or quadratic surds. For example√2, √3, √5, √7, √x are the surds of order 2.

Example: √2, √5, √10, √a, √m, √x, √(x + 1) are second order surd or quadratic surd (since the indices of roots are 2).

(ii) A surd with index of root 3 is called a third order surd or cubic surd.

If x is a positive integer with nth root, then  is a surd of nth order when the value of  is irrational. In  expression n is the order of surd and x is called as radicand. For example  is surd of order 3.

The surds which have the indices of cube roots are called as third order surds or cubic surds. For example ∛2, ∛3, ∛10, ∛17, ∛x are the surds of order 3 or cubic surds.

Example: ∛2, ∛5, ∛7, ∛15, ∛100, ∛a, ∛m, ∛x, ∛(x - 1) are third order surd or cubic surd (since the indices of roots are 3).


(iii) A surd with index of root 4 is called a fourth order surd.

The surds which have the indices of four roots are called as forth order surds or bi-quadratic surds.

For example ∜2, ∜4, ∜9, ∜20, ∜x are the surds of order 4.

Example: 42, 43, 49, 417, 470, 4a, 4m, 4x, 4x1 are third order surd or cubic surd (since the indices of roots are 4).


(iv) In general, a surd with index of root n is called a nth order surd.

Similarly the surds which have the indices of n roots are nth order surds. n2, n17, n19, nx are the surds of order n.

Example: n2, n3, n9, n17, n70, na, nm, nx, nx1 are nth order surd (since the indices of roots are n).


Problem on finding the order of a surd:

Express ∛4 as a surd of order 12.

Solution:

Now, ∛4

= 41/3

= 41×43×4, [Since, we are to convert order 3 into 12, so we multiply both numerator and denominator of 1/3 by 4]

= 44/12

= 1244

= 12256


Problems on finding the order of surds:

1. Express √2 as a surd of order 6.

Solution:

√2 = 21/2

     = 21×32×3

     = 236

     = 81/6

     = 68

So 68 is a surd of order 6.


2. Express ∛3 as a surd of order 9.

Solution:

∛3 = 31/3

     = 31×33×3

     = 339

     = 271/9

     = 927

So 927 is a surd of order 9.


3. Simplify the surd  ∜25 to a quadratic surd.

Solution:

 ∜25 = 251/4

= 52×14

= 312

= 25

= √5

So √5 is a surd of order 2 or a quadratic surd.














11 and 12 Grade Math

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