Exponential Form: Exponentiation is a mathematical operation, 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}


\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