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.

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