**Order of a Surd**

**Definitions of surds**: A root of a positive real quantity is called a surd if its value cannot be exactly determined. It is a number that can’t be simplified to remove a square root (or cube root etc). For example, each of the quantities √3, ∛7, ∜19, (16)^2/5 etc. is a surd.

More Examples:

- √2 (square root of 2) can’t be simplified further so it is a surd
- √4 (square root of 4) CAN be simplified to 2, so it is NOT a surd

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

In ^{a}√n, n is called the order of the surd and a is called the radicand.

For example, The order of the surd ^{5}√z is 5.

(i) A surd with an 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 an 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).

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

Example: ^{4}√2, ^{4}√3, 4√9, ^{4}√17, ^{4}√70, ^{4}√a, ^{4}√m, ^{4}√x, ^{4}√(x-1) are third order surd or cubic surd (since the indices of roots are 4).

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

Example: ^{n}√2, ^{n}√3, ^{n}√9, ^{n}√17, ^{n}√70, ^{n}√a, ^{n}√m, ^{n}√x, ^{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}

= 44^(1×4)/(3×4), [Since we are to convert order 3 into 12, so we multiply both a numerator and denominator of 1/3 by 4]

= 4^{4/12 }= ^{12}√4^{4 }= ^{12}√256.

Information Source: