# 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.

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.

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).

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

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).

(v) In general, a surd with index of root n is called a n$$^{th}$$ order surd.

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

= 4$$^{4/12}$$

= $$\sqrt[12]{4^{4}}$$

= $$\sqrt[12]{256}$$