Surds are irrational roots of a rational number.

e.g. \(\sqrt{6} =\) a surd ⇒ it can’t be exactly found. Similarly \(- \sqrt{7}, \sqrt{8}, \sqrt[3]{9}, \sqrt[4]{27}\) etc. are all surds.

Pure Surd : The surds which are made up of only an irrational number e.g. \(√6, √7, √8\) etc.
Mixed Surd : Surds which are made up of partly rational and partly irrational numbers e.g. \(3√3, 6^4√27\) etc.

Q. Convert \(√27\) to a mixed surd A. \(√27 = √9 \times 3 = 3√3\)
Q. Convert \(2√8\) to a pure surd A. \(2√8 = √8 \times 4 = √32\)

Rationalization of Surds: In order to rationalize a given surd, multiply and divide by the conjugate of denominator [conjugate of \((a + √b)\) is \((a – √b)\) and vice versa].

e.g. \(\frac{(6+\sqrt{2})}{(1-\sqrt{3})} = \frac{(6+\sqrt{2})(1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})}= \frac{(6+6 \sqrt{3} +\sqrt{2}+\sqrt{6})}{(1-3)}=\frac{(6+6 \sqrt{3 }+\sqrt{2} +\sqrt{6})}{-2}\)