Surd : Let a be a rartional number and m be a postive integer such that  $ a^{\cfrac{1}{m}} = {\sqrt[m]{a}} $ is irrational. Then $  \sqrt[m]{a}  $ is called a surd of order n.

Examples :

• ${\sqrt[2]{5}} $ is a surd of order 2.
• ${\sqrt[3]{7}} $ is a surd of order 3.
• $4{\sqrt[2]{7}} $ is a surd of order 2.

Note :

1. Note that numbers like ${\sqrt[2]{16}},{\sqrt[3]{125}}$, etc. are not surds because they are not irrational numbers.
2. Every surd is an irrational number. But every irrational number is not a surd. (eg : π , e, etc. are not surds though they are irrational numbers.)

1. $ a^m \times a^n =a^{m+n} $
2. $ \cfrac{a^m}{a^n}=a^{m-n} $
3. $ (a^{m})^{n}=a^{mn} $
4. $ (ab)^n=a^{n}{b^n} $
5. $\left( {\cfrac{a}{b}}\right)^n=\cfrac{a^n}{b^n}$
6. $ a^0=1 $

1. $ {\sqrt[n]{a}}=a^{\cfrac{1}{n}} $
2. $ \sqrt[n]{ab}=\sqrt[n]{a} \times \sqrt[n]{b} $
3. $ \sqrt[n]{\cfrac{a}{b}} =\cfrac{\sqrt[n]{a}}{\sqrt[n]{b}} $
4. $ {(\sqrt[n]{a})}^n=a $
5. $ \sqrt[m]{\sqrt[n]{a}} =\sqrt[mn]{a} $
6. $ {(\sqrt[n]{a})^m} =\sqrt[n]{a^m} $
7. $ \sqrt{a\cdot \sqrt{a\cdot\sqrt{a\cdot\sqrt{a\cdot\cdot \cdot n\ times}}}}$ $ = a^{1-\cfrac{1}{2^n}} $
8. $ \sqrt{a\cdot \sqrt{a\cdot\sqrt{a\cdot\sqrt{a\cdot\cdot \cdot \infty}}}}$ $= a$