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}

Note:

  1. If \frac{p}{q} is a rational number, then (\frac{p}{q})^m = \frac{p^m}{q^m}
  2. Reciprocal of \frac{p}{q} is \frac{q}{p} , where p\neq 0  and q\neq 0
  3. Let x be any non-zero real number and m and n be positive integers:

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

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

(x^m)^n=x^{mn}

x^0=1

(xy)^m=x^my^n

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

x^{-m} = \frac{1}{x^m}   and \frac{1}{x^{-m}} = x^m

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

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