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The properties of whole number exponents also hold for negative exponents. This table contains an example of each property.

 Property Positive Integer Exponents Negative Integer Exponents Multiplication 32 · 34 = 32 + 4 = 36 Division  Power of a Power (52)3 = 52 · 3 = 56 (5 -2) -3 = 5(-2) · (-3) = 56 Power of a Product (5 · 7)3 = 53 · 73 Power of a Quotient  Now we will find two additional properties of negative exponents.

 Well begin by simplifying  We apply the definition of a negative exponent. Rewrite the division using ÷. To divide by a fraction, multiply by its reciprocal. Multiply the numerators. Multiply the denominators. Thus, Notice that the bases, 2 and 5, have moved to the opposite side of the division bar, and the signs of the their exponents changed.

This turns out to be true in general.

Next, well use this relationship to rewrite a quotient raised to a negative power.

 For example, well simplify  We use the Power of a Quotient Property. As in the previous example, we move each base to the opposite side of the division bar and change the sign of each exponent. Again, we use the Power of a Quotient Property. We see that Notice that the new base, , is the reciprocal of the original base. Also notice the new exponent, 2, is the opposite of the original exponent.

Note:

To rewrite a fraction raised to a negative power, just flip the fraction and change the negative power to positive. 