Zero Exponent
We have used positive and negative integral exponents, but we have not yet seen the
integer 0 used as an exponent. Note that the product rule was stated to hold for any
integers m and n. If we use the product rule on 2^{3} Â· 2^{3}, we get
2^{3} Â· 2^{3} = 2^{0}.
Because 2^{3} 8 and
, we must have 2^{3}
Â· 2^{3} = 1. So for consistency we
define 2^{0} and the zero power of any nonzero number to be 1.
Zero Exponent
If a is any nonzero real number, then a^{0} = 1.
Example
Using zero as an exponent
Simplify each expression. Write answers with positive exponents and assume all
variables represent nonzero real numbers.
Solution
a) To evaluate 3^{0}, we find 3^{0} and then take the opposite. So
3^{0} = 1.
b)
Definition of zero exponent
c) 2a^{5}b^{6} Â· 3a^{5}b^{2}

= 6a^{5} Â· a^{5} Â· b^{6}
Â· b^{2} = 6a^{0}b^{4}

Product rule
Definitions of negative and zero exponent

Helpful Hint
Defining a^{0} to be 1 gives a consistent
pattern to exponents:
If the exponent is increased by
1 (with base 3) the value of the
expression is multiplied by 3.
