8.5: Negative Exponents
What if you had a mathematical expression like
Watch This
Foundation: 0805S Negative Exponents
Guidance
The product and quotient rules for exponents lead to many interesting concepts. For example, so far we’ve mostly just considered positive, whole numbers as exponents, but you might be wondering what happens when the exponent isn’t a positive whole number. What does it mean to raise something to the power of zero, or 1, or
Simplify Expressions With Negative Exponents
When we learned the quotient rule for exponents
Negative Power Rule for Exponents:
Negative exponents can be applied to products and quotients also. Here’s an example of a negative exponent being applied to a product:
And here’s one applied to a quotient:
That last step wasn’t really necessary, but putting the answer in that form shows us something useful:
Negative Power Rule for Fractions:
This rule can be useful when you want to write out an expression without using fractions.
Example A
Write the following expressions without fractions.
a)
b)
c)
d)
Solution
a)
b)
c)
d)
Example B
Simplify the following expressions and write them without fractions.
a)
b)
Solution
a) Reduce the numbers and apply the quotient rule to each variable separately:
b) Apply the power rule for quotients first:
Then simplify the numbers, and use the product rule on the
You can also use the negative power rule the other way around if you want to write an expression without negative exponents.
Example C
Write the following expressions without negative exponents.
a)
b)
c)
d)
Solution
a)
b)
c)
d)
Watch this video for help with the Examples above.
CK12 Foundation: Negative Exponents
Vocabulary

Negative Power Rule for Exponents:
x−n=1xn , wherex≠0.
Guided Practice
Simplify the following expressions and write the answers without negative powers.
a)
b)
Solution
a) Apply the quotient rule inside the parentheses:
Then apply the power rule:
b) Apply the quotient rule to each variable separately:
Practice
Simplify the following expressions in such a way that there aren't any negative exponents in the answer.

x−1y2 
x−4 
x−3x−7 
x−3y−5z−7 
(ab)−2 
(3a−2b2c3)3
Simplify the following expressions in such a way that there aren't any fractions in the answer.

a−3(a5)a−6 
5x6y2x8y 
(4ab6)3(ab)5 
(3x3)(4x4)(2y)2 
a−2b−3c−1
Negative Exponent Property
The negative exponent property states that and for .quotient rule
In calculus, the quotient rule states that if and are differentiable functions at and , then .Zero Exponent Property
The zero exponent property says that for all , .Image Attributions
Description
Learning Objectives
Here you'll learn how to simplify expressions that contain negative exponents.