### Theory:

Surds can be classified into the following types. They are:

1. Surds of same order: Surds having the same root of index are called as surds of same order. They are also called as equiradical surds.
Example:
$$\sqrt[3]{4}$$, $$\sqrt[3]{7}$$ and $$\sqrt[3]{13}$$ are surds of same order having root index $$3$$.
2. Simplest form of a surd: When a surd is simplified and expressed as a product of rational and an irrational number, then the surd is said to be in simplest form. In the simplest form, the surd has
• the smallest index of the radical.
• the radical sign will have no fraction.
• there will be no term of the form $$a^n$$, where $$a$$ is a positive integer under index $$n$$.
Example:
Write the surds $$\sqrt{12}$$ and $$\sqrt[4]{64}$$ in simplest form.

Solution:

$$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$

$$\sqrt[4]{64} = \sqrt[4]{2 \times 2 \times 2 \times 2 \times 4} = 4\sqrt[4]{4}$$
3. Pure and Mixed surds: If the coefficient of a surd is $$1$$, then the surd is said to be a pure surd. If the coefficient of a surd having other than $$1$$ is called as mixed surd.
Example:
$$\sqrt[3]{5}$$, $$\sqrt{2}$$, $$\sqrt[4]{7}$$ are pure surds.

$$2\sqrt[5]{16}$$, $$3\sqrt[4]{9}$$, $$7\sqrt[3]{5}$$ are mixed surds.
4. Simple and compound surds: If a surd has only $$1$$ term, then the surd is said to be a simple surd. The sum or difference of $$2$$ or more surds is called as compound surd.
Example:
$$\sqrt{3}$$, $$4 \sqrt[3]{2}$$ are simple surds.

$$\sqrt[3]{5} + \sqrt[4]{13}$$, $$\sqrt{5} - 7\sqrt[3]{8}$$ are compound surds.
5. Binomial surd: The sum or difference of $$2$$ terms which consists of either $$2$$ surds or $$1$$ is a rational number and an another is an irrational number is called as a binomial surd.
Example:
$$\sqrt{7} - 6 \sqrt[3]{5}$$, $$\frac{2}{7} + \sqrt[4]{13}$$ are binomial surds.