8.3: Zero, Negative, and Fractional Exponents
In the previous lessons, we have dealt with powers that are positive whole numbers. In this lesson, you will learn how to solve expressions when the exponent is zero, negative, or a fractional number.
Exponents of Zero: For all real numbers
Example:
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:
Negative Power Rule for Exponents:
Example:
Multimedia Link: For more 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.
Example 1: Write the following expressions without fractions.
(a)
(b)
Solution:
(a)
(b)
Notice in Example 1(a), the number 2 is in the numerator. This number is multiplied with
Simplifying Expressions with Fractional Exponents
The next objective is to be able to use fractions as exponents in an expression.
Roots as Fractional Exponents:
Example:
Example 2: Simplify the following expressions.
(a)
(b)
Solution:
(a)
(b)
It is important when evaluating expressions that you remember the Order of Operations. Evaluate what is inside the parentheses, then evaluate the exponents, then perform multiplication/division from left to right, then perform addition/subtraction from left to right.
Example 3: Evaluate the following expression.
(a)
Solution:
Practice Set
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 
a12⋅a13 
(a13)2 
a52a12 
(x2y3)13 
x−3y−5z−7 
(x12y−23)(x2y13) 
(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 
(3xy13)3 
4a2b32a5b 
(x3y2)3⋅x2y4 
(ab−2b3)2 
x−3y2x2y−2 
3x2y32xy12 
(3x3)(4x4)(2y)2 
a−2b−3c−1 
x12y52x32y32
Evaluate the following expressions to a single number.

3−2 
(6.2)0 
8−4⋅86 
(1612)3 
50 
72 
(23)3 
3−3 
1612 
8−13
In 43 – 45, evaluate the expression for

2x2−3y3+4z 
(x2−y2)2 
(3x2y54z)−2  Evaluate
x24x3y44y2 ifx=2 andy=−1 .  Evaluate
a4(b2)3+2ab ifa=−2 andb=1 .  Evaluate
5x2−2y3+3z ifx=3, y=2, andz=4 .  Evaluate
(a2b3)−2 ifa=5 andb=3 .  Evaluate
3⋅55−10⋅5+1 .  Evaluate
2⋅42−3⋅5232 .  Evaluate
(3322)−2⋅34 .
Mixed Review
 A quiz has ten questions: 7 true/false and 3 multiple choice. The multiple choice questions each have four options. How many ways can the test be answered?
 Simplify
3a4b4⋅a−3b−4 .  Simplify
(x4y2⋅xy0)5 .  Simplify
v2−vu−2⋅u−1v4 .  Solve for
n:−6(4n+3)=n+32 .