Order of a Surd

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

In \(\sqrt[n]{a}\), n is called the order of the surd and a is called the radicand.

For example: The order of the surd \(\sqrt[5]{z}\) 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: \(\sqrt[4]{2}\), \(\sqrt[4]{3}\), \(\sqrt[4]{9}\), \(\sqrt[4]{17}\), \(\sqrt[4]{70}\), \(\sqrt[4]{a}\), \(\sqrt[4]{m}\), \(\sqrt[4]{x}\), \(\sqrt[4]{x - 1}\) 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 n\(^{th}\) order surd.

Similarly the surds which have the indices of n roots are nth order surds. \(\sqrt[n]{2}\), \(\sqrt[n]{17}\), \(\sqrt[n]{19}\), \(\sqrt[n]{x}\) are the surds of order n.

Example: \(\sqrt[n]{2}\), \(\sqrt[n]{3}\), \(\sqrt[n]{9}\), \(\sqrt[n]{17}\), \(\sqrt[n]{70}\), \(\sqrt[n]{a}\), \(\sqrt[n]{m}\), \(\sqrt[n]{x}\), \(\sqrt[n]{x - 1}\) 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

= 4\(^{1/3}\)

= \(4^{\frac{1 × 4}{3 × 4}}\), [Since, we are to convert order 3 into 12, so we multiply both numerator and denominator of 1/3 by 4]

= 4\(^{4/12}\)

= \(\sqrt[12]{4^{4}}\)

= \(\sqrt[12]{256}\)


Problems on finding the order of surds:

1. Express √2 as a surd of order 6.

Solution:

√2 = 2\(^{1/2}\)

     = \(2^{\frac{1 × 3}{2 × 3}}\)

     = \(2^{\frac{3}{6}}\)

     = 8\(^{1/6}\)

     = \(\sqrt[6]{8}\)

So \(\sqrt[6]{8}\) is a surd of order 6.


2. Express ∛3 as a surd of order 9.

Solution:

∛3 = 3\(^{1/3}\)

     = \(3^{\frac{1 × 3}{3 × 3}}\)

     = \(3^{\frac{3}{9}}\)

     = 27\(^{1/9}\)

     = \(\sqrt[9]{27}\)

So \(\sqrt[9]{27}\) is a surd of order 9.


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

Solution:

 ∜25 = 25\(^{1/4}\)

= \(5^{\frac{2 × 1}{4}}\)

= \(3^{\frac{1}{2}}\)

= \(\sqrt[2]{5}\)

= √5

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














11 and 12 Grade Math

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