Product Rule for Exponents
We can simplify an expression such as 23 · 25 using the definition of exponents.
Notice that the exponent 8 is the sum of the exponents 3 and 5. This example illustrates
the product rule for exponents.
Product Rule for Exponents
If m and n are integers and a ≠ 0, then
am · an = am + n.
Example
Using the product rule
Simplify each expression. Write answers with positive exponents and assume all
variables represent nonzero real numbers.
a) 34 · 36
b) 4x-3 · 5x
c) -2y-3(-5y-4)
Solution
| a) 34 · 36 = 34 + 6 = 310
|
Product rule |
| b) 4x-3 · 5x |
= 4 · 5 · x-3 · x1 |
|
| |
= 20x-2 |
Product rule: x-3 · x1 = x-3 +
1 = x-2 |
| |
 |
Definition of negative exponent |
| c) -2y-3(-5y-4) |
= (-2)(-5)y-3y-4 |
|
| |
= 10y-7 |
Product rule: -3 + (-4) = -7 |
| |
 |
Definition of negative exponent |
Caution
The product rule cannot be applied to 23 · 32 because the bases
are not identical. Even when the bases are identical, we do not multiply the bases.
For example, 25 · 24 ≠ 49. Using the rule correctly, we get 25
· 24 = 29.
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