Properties of Negative Exponents

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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 = 3^{2 + 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.

Well 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, well use this relationship to rewrite a quotient raised to a negative power.

For example, well 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.