Quotient Rule for Exponents
We can use arithmetic to simplify the quotient of two exponential expressions. For
example,
![](./articles_imgs/914/quotie49.gif)
There are five 2’s in the numerator and three 2’s in the denominator. After dividing,
two 2’s remain. The exponent in 22 can be obtained by subtracting the
exponents 3 and 5. This example illustrates the quotient rule for exponents.
Quotient Rule for Exponents
If m and n are any integers and a ≠ 0, then
![](./articles_imgs/914/quotie50.gif)
Example 1
Using the quotient rule
Simplify each expression. Write answers with positive exponents only. All variables
represent nonzero real numbers.
![](./articles_imgs/914/quotie51.gif)
Solution
![](./articles_imgs/914/quotie52.gif)
The next example further illustrates the rules of exponents. Remember that the
bases must be identical for the quotient rule or the product rule.
Example 2
Using the product and quotient rules
Use the rules of exponents to simplify each expression. Write answers with positive
exponents only. All variables represent nonzero real numbers.
![](./articles_imgs/914/quotie53.gif)
Solution
![](./articles_imgs/914/quotie54.gif) |
= 2x0 |
Quotient rule: -7 - (-7) = 0 |
|
= 2 |
Definition of zero exponent |
![](./articles_imgs/914/quotie55.gif) |
![](./articles_imgs/914/quotie56.gif) |
Product rule: w1 · w-4 = w-3 |
|
![](./articles_imgs/914/quotie57.gif) |
Quotient rule: -3 - (-2) = -1 |
|
![](./articles_imgs/914/quotie58.gif) |
Definition of negative exponent |
|