## Surds and Indices

### Laws of Indices

Laws of Indices:

• $$a^{m} \times a^{n}=a^{m+n}$$
• $$\frac{a^{m}}{a^{n}}=a^{m-n}$$
• $$(a^{m})^n=a^{m n}$$
• $$(a b)^{n}=a^{n} b^{n}$$
• $$\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}$$
• $$a^{0}=1$$

### SURDS

Surds: Let a be a rational number and n  be a positive  integer such that $$a^{\frac{1}{n}}=\sqrt[5]{a}$$  is irrational. Then, $$\sqrt[n]{a}$$  is called a surd of order n.

LAWS OF SURDS:

1. $$\sqrt[n]{a}=a^{\frac{1}{n}}$$
2. $$\sqrt[n]{a b}=\sqrt[n]{a} \times \sqrt[n]{b}$$
3. $$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$
4. $$\sqrt[m]{\sqrt[n]{\alpha}}=\sqrt[m n]{a}$$
5. $$(\sqrt[n]{a})^{n}=a$$
6. $$(\sqrt[n]{a})^{m}=\sqrt[n]{a^{m}}$$