8.3: Negative Exponents
All the students in a class were randomly given an expression, and they were asked to make pairs, with one boy and one girl per pair. The students were asked to divide the boy's expression by the girl's expression. Bill and Jenna paired up, with Bill having
Watch This
Multimedia Link: For help with these types of exponents, watch this http://www.phschool.com/atschool/academy123/english/academy123_content/wlbookdemo/ph241s.html  PH School video or visit the http://www.mathsisfun.com/algebra/negativeexponents.html  mathisfun website.
Guidance
In the previous Concepts, we have dealt with powers that are positive whole numbers. In this Concept, you will learn how to solve expressions when the exponent is zero or a negative number.
Exponents of Zero: For all real numbers
Example A
Simplify
Solution:
Simplifying Expressions with Negative Exponents
The next objective is negative exponents. When we use the quotient rule and we subtract a greater number from a smaller number, the answer will become negative. The variable and the power will be moved to the denominator of a fraction. You will learn how to write this in an expression.
Example B
Simplify
Solution:
Negative Power Rule for Exponents:
Example C
Rewrite using only positive exponents:
Solution:
Example D
Write the following expressions without fractions.
(a)
(b)
Solution:
(a)
(b)
Notice in part (a), the number 2 is in the numerator. This number is multiplied with
Guided Practice
Simplify
Solution:
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Zero, Negative, and Fractional Exponents (14:04)
Simplify the following expressions. Be sure the final answer includes only positive exponents.

x−1⋅y2 
x−4 
x−3x−7 
1x 
2x2 
x2y3 
3xy 
3x−3 
a2b−3c−1 
4x−1y3 
2x−2y−3 
(ab)−2 
(3a−2b2c3)3 
x−3⋅x3
Simplify the following expressions without any fractions in the answer.

a−3(a5)a−6 
5x6y2x8y 
(4ab6)3(ab)5
Evaluate the following expressions to a single number.

3−2 
(6.2)0 
8−4⋅86
In 21 – 23, evaluate the expression for

2x2−3y3+4z 
(x2−y2)2 
(3x2y54z)−2  Evaluate \begin{align*}x^24x^3y^44y^2\end{align*} if \begin{align*}x=2\end{align*} and \begin{align*}y=1\end{align*}.
 Evaluate \begin{align*}a^4(b^2)^3+2ab\end{align*} if \begin{align*}a=2\end{align*} and \begin{align*}b=1\end{align*}.
 Evaluate \begin{align*}5x^22y^3+3z\end{align*} if \begin{align*}x=3, \ y=2,\end{align*} and \begin{align*}z=4\end{align*}.
 Evaluate \begin{align*}\left(\frac{a^2}{b^3}\right)^{2}\end{align*} if \begin{align*}a=5\end{align*} and \begin{align*}b=3\end{align*}.
 Evaluate \begin{align*}3 \cdot 5^5  10 \cdot 5+1\end{align*}.
 Evaluate \begin{align*}\frac{2 \cdot 4^23 \cdot 5^2}{3^2}\end{align*}.
 Evaluate \begin{align*}\left(\frac{3^3}{2^2}\right)^{2} \cdot \frac{3}{4}\end{align*}.
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exponents of zero
For all real numbers .Negative Power Rule for Exponents
where .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
Here you'll learn how to use zero and negative exponents to simplify expressions.