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# Zero Power Property of Exponents

Property â€” Zero Power Property

English Any real number, except zero, raised to the power 0 is 1.

Algebra x0 = 1, x 0

Example 170 = 1

Hereâ€™s a way to understand why 170 is 1.

Suppose we write 0 as 2 - 2. Then, 170 = 172 - 2.

By the Division Property of Exponents, Since 170 = 172 - 2 and 172 - 2 = 1, we have 170 = 1.

Note:

This same reasoning applies no matter what power or nonzero base we choose. Therefore, x0 = 1 for x 0.

Example 1

a. Use the Zero Power Property to simplify 50.

Solution

 a. Any real number, except zero, raised to the power 0 is 1. 50 = 1 b. Suppose we have We can simplify this using the Division Property of Exponents. But if we reduce the fraction the result is 1. Since is equivalent to both 50 and 1, we conclude 50 = 1. Example 2

Find each of the following. (Assume each variable represents a nonzero real number).

 a. (-7)0 b. c. (12x4y5)0 d. -2y0 e. 00

Solution

In each case, we apply the Zero Power Property: any nonzero real number raised to the zero power is 1.
 a. The base is the real number -7. (-7)0 = 1 b. The base, w, represents a nonzero real number.  c. The base, 12x4y5, represents a nonzero real number. (12x4y5)0 = 1 d. Only y is raised to the power 0. -2y0 = -2 Â· 1 = -2 e. In the Zero Power Property, the base cannot be 0. 00 is undefined