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Negative Exponents

Negative exponents mean that instead of multiplying that many of the base number together, you divide.

For example, 5 2 = 25 but 5 −2 = 1 5 2 = 1 25 .

You can use the product of powers property to show this.

5 −2 × 5 2 = 5 ( −2 + 2 ) = 5 0

We know 5 2 = 25 , and we know 5 0 = 1 . So, this says that 5 −2 × 25 = 1 . What number times 25 equals 1 ? That would be its multiplicative inverse , 1 25 .

5 −2 = 1 25

In general, for all real numbers a and b , where a 0 , a − b is defined to be 1 a b , so

a − b = 1 a b