The Product and Quotient Rules
The rules that we have already discussed are summarized below.
The following rules hold for nonnegative integers m and n and a
≠ 0.
a m · a n = a m + n |
Product rule |
![](./articles_imgs/871/prod1.gif) |
Quotient rule |
![](./articles_imgs/871/prod2.gif) |
a 0 = 1 |
Zero exponent |
Cuation
The product and quotient rules apply only if the bases of the expressions are
identical. For example, 32 · 34 = 36,
but the product rule cannot be applied to 52 · 34. Note
also that the bases are not multiplied: 32 · 34
≠ 96.
Note that in the quotient rule the exponents are
always subtracted, as in
![](./articles_imgs/871/prod3.gif)
If the larger exponent is in the denominator,
then the result is placed in the denominator.
Example 1
Using the product and quotient rules
Use the rules of exponents to simplify each
expression. Assume that all variables represent nonzero real numbers.
![](./articles_imgs/871/prod4.gif)
Solution
a) Because the bases are both 2, we can use the
product rule:
23 · 22 |
= 25 |
Product rule |
|
= 32 |
Simplify |
b) (3x)0(5x2)(4x) |
= 1 · 5x2 · 4x |
Definition of zero exponent |
|
= 20x3 |
Product rule |
c)
![](./articles_imgs/871/prod5.gif) |
Quotient rule |
d) First use the product rule to simplify the numerator and denominator:
![](./articles_imgs/871/prod6.gif) |
Product rule |
Quotient rule |
|