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 Depdendent Variable

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 Dependent Variable

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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 Â· 2-3, we get 23 Â· 2-3 = 20.

Because 23  8 and , we must have 23 Â· 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 Â· 3a-5b2 = -6a5 Â· a-5 Â· b-6 Â· b2= -6a0b-4 Product rule Definitions of negative and zero exponent