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.