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.