In a very simple way, A surd is a square root which cannot be reduced to a whole number. For example, \sqrt{4}=2  is not a surd, as the answer is a whole number. But \sqrt{5}  is not a whole number. You could use a calculator to find that \sqrt{5} = 2.236067977... but instead of this we often leave our answers in the square root form, as a surd.

Another definition: When we can’t simplify a number to remove a square root (or cube root etc) then it is a surd. 

\displaystyle \text{ Positive }  n^{th} \text{ Root: The positive } n^{th} \text{ root of a is denoted by }  \sqrt[n]{a}  \text{ or }  a^{\frac{1}{n}}

Formal Definition: If a is a positive rational number and n is a positive integer, such that the n^{th} root of a i.e. \sqrt[n]{a} or a^{\frac{1}{n}} is an irrational number, then \sqrt[n]{a} or a^{\frac{1}{n}} is called a surd.

Examples:

  • \sqrt{5} or 5^{\frac{1}{2}} is a surd of order 2
  • \sqrt[3]{8} or 8^{\frac{1}{3}} is not a surd because it is equal to 2 which is not an irrational number.
  • \sqrt{\sqrt{\sqrt{2}}} = \sqrt[8]{2} . Clearly 2 is a positive rational number such that \sqrt[8]{2} is an irrational number. Therefore \sqrt[8]{2} is a surd of the order 8
  • Similarly, \sqrt{\sqrt[3]{729}} = 3 . Therefore \sqrt{\sqrt[3]{729}} is not a surd.

Type of Surds

Quadratic Surds: Surds of order 2 .

Examples: \sqrt{2}, \sqrt{3}, \sqrt{5}, etc.

Cubic Surds: Surds of order 3 .

Examples: \sqrt[3]{2}, \sqrt[3]{3}, \sqrt[3]{5}, etc.

Bioquadratic Surds: Surds of order 4 .

Examples: \sqrt[4]{2}, \sqrt[4]{3}, \sqrt[4]{5}, etc.

Pure Surds: A surd having no rational factor other than unity is called a pure surd.

Examples: \sqrt{3}, \sqrt[3]{2}, \sqrt{5}

Mixed Surds: Surds having a rational factor other than unity is called mixed surd. Example

Laws of Surds:

First Law: If a is a positive rational number and n is a positive number then \displaystyle (\sqrt[n]{a})^n = a \ or \  (a^n)^{\frac{1}{n}} = a 

Second Law: If a, b is a positive rational number and n is a positive number then \displaystyle \sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{an}

Third Law: If a, b is a positive rational number and n is a positive number then \displaystyle \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}

Fourth Law: If a is a positive rational number and m, n are  positive numbers then \displaystyle \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} = \sqrt[n]{\sqrt[m]{a}}

Fifth Law: If a is a positive rational number and m, n are  positive numbers then \displaystyle \sqrt[n]{\sqrt[m]{(a^p)^m}} = \sqrt[n]{a^p} = \sqrt[mn]{a^{pm}}

Another way of classifying surds:

Monomial Surds: A surd containing only one term is called a monomial surd.

Example: \sqrt{2}, \sqrt{3}, \sqrt[5]{4}

Binomial Surds: An expression consisting of the sum or difference of two monomial surds or sum of a monomial surd and a rational number is called a binomial surd.

Examples: 2 + \sqrt{3}, 2 - \sqrt{3}, \sqrt{3}-2, \sqrt{3}+\sqrt{5}, \sqrt{5}-\sqrt{3}

Trinomial Surds: An expression consisting of three terms of which at least two are monomial surds is called a trinomial surd.

Examples: \sqrt{2}+\sqrt{3}+\sqrt{5}, 2 + \sqrt{3}+\sqrt{7}, \sqrt{17}-\sqrt{2}- 2

Rationalising Factor: If the product of two surds is a rational number, then each one of them is called the rationalizing factor of the other.

Example: \sqrt{3} is a rationalising factor of 2\sqrt{3} because \sqrt{3} \times 2\sqrt{3} = 6 which is a rational number.

Similarly, (\sqrt{3}-\sqrt{2}) is a rationalising factor of (\sqrt{3}+\sqrt{2}) because (\sqrt{3}-\sqrt{2}) \times (\sqrt{3}+\sqrt{2}) = 3-2 = 1 which is a rational number.

Conjugate Surds: Two binomial surds which differ only in sign (+ \ or \ -) between the terms connecting them are called conjugate surds. (a+\sqrt{b}) and (a-\sqrt{b}) are conjugate surds. Also (\sqrt{a}+\sqrt{b}) and (\sqrt{a}-\sqrt{b}) are conjugate surds.