Exponential Form: Exponentiation is a , written as $b^n$, involving two numbers, the base $b$ and the exponent $n$. When $n$ is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is $b^n$, is the product of multiplying $n$ bases:

$b^n=b\times b\times b\times ... \times b$

In that case,$b^n$, is called the $n^{th}$ power of $b$, or $b$ raised to the power $n$.

When $n$ is a negative integer and $b$ is not zero,  $b^{-n}$  is naturally defined as $\frac{1}{b^n}$

Note:

$\displaystyle \text{1. If } \frac{p}{q} \text{ is a rational number, then } (\frac{p}{q})^m = \frac{p^m}{q^m}$

$\displaystyle \text{2. Reciprocal of } \frac{p}{q} \text{ is } \frac{q}{p} \text{ , where } p\neq 0 \text{ and } q\neq 0$

Let $x$ be any non-zero real number and $m$ and $n$ be positive integers:

$x^m\times x^n= x^{m+n}$

$\displaystyle \frac{x^m}{x^n} =x^{m-n} \text{ , where } m>n \text{ and } \frac{x^m}{x^n} = \frac{1}{x^{n-m}} \text{ , where } n>m$

$\displaystyle (x^m)^n=x^{mn}$

$\displaystyle x^0=1$

$\displaystyle (xy)^m=x^my^n$

$\displaystyle (\frac{x}{y})^m = \frac{x^m}{y^m}$

$\displaystyle x^{-m} = \frac{1}{x^m} \text{ and } \frac{1}{x^{-m}} = x^m$

If  $\displaystyle x^m=x^n$,  then $m=n$,  provided $x>0$ and $x\neq 1$

$\displaystyle (x)^{\frac{m}{n}}=(x^m)^{\frac{1}{n}}=(x^{\frac{1}{n}})^m$