Rules Of Indices

Introduction

Four important rules of indices are:

• Rule of Multiplication
• Rule of Division
• Rule of Power of a Power
• Rule of Powers of Brackets

Four important rules of indices are listed in table 1. Rules of multiplication and division are applied for indices containing identical base.

No.	Rules of indices	Examples
1	Rule of Multiplication $a^m\times a^n=a^{m+n}$	$$\begin{array}{l}{2}^3\times 2^4=2^{3+4}\\ =2^7\end{array}$$
2	Rule of Division $a^m÷a^n=a^{m-n}$	$$\begin{array}{l}{3}^4÷3^2=3^{4-2}\\ =3^2\end{array}$$
3	Rule for power of a power ${\left(a^m\right)}^n=a^{m\times n}$	$$\begin{array}{l}{\left(4^2\right)}^3=4^{2\times 3}\\ =4^6\end{array}$$
4	Rule of Powers of Brackets $${\left(\frac{a^m{b}^n}{c^p}\right)}^r=\frac{a^{m\times r}{b}^{n\times r}}{c^{p\times r}}$$	$$\begin{array}{l}{\left(\frac{3ab}{2c^2}\right)}^2=\frac{3^2{a}^2{b}^2}{2^2{c}^{2\times 2}}\\ =\frac{9a^2{b}^2}{4c^4}\end{array}$$

Other important rules - applied for indices containing different base.

No.	Rules of indices	Examples
1	Different base but similar power $$\begin{array}{l}{a}^m\times b^m={\left(a\times b\right)}^m\\ c^n÷d^n={\left(\frac{c}{d}\right)}^n\end{array}$$	$$\begin{array}{l}{e}^3\times f^3={\left(e\times f\right)}^3\\ ={\left(ef\right)}^3\end{array}$$
2	Zero Power $$a^0=1$$	$$6^0=1$$
3	Negative Power $$\frac{1}{a^m}=a^{-m}$$	$$\frac{1}{4^2}=4^{-2}$$
4	Rule of Root $$\sqrt[m]{a^n}=a^{\frac{n}{m}}$$	$$\sqrt[3]{a^2}=a^{\frac{2}{3}}$$