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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 23 · 2-3, we get 23 · 2-3 = 20.

Because 23  8 and , we must have 23 · 2-3 = 1. So for consistency we define 20 and the zero power of any nonzero number to be 1.

Zero Exponent

If a is any nonzero real number, then a0 = 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 -30, we find 30 and then take the opposite. So -30 = -1.

b) Definition of zero exponent

c) -2a5b-6 · 3a-5b2

 

 

= -6a5 · a-5 · b-6 · b2

= -6a0b-4

 

Product rule

Definitions of negative and zero exponent

 

Helpful Hint

Defining a0 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.

 
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