Notes: Simplifying Integer Exponents
Name:
A numeric expression is not simplified if there is ANY exponent still in the expression, except 1.
Examples
NOT SIMPLIFIED EXAMPLES
SIMPLIFIED EXAMPLES
3
17
2
-15
4−5
225
70
An algebraic expression is not simplified if there are negative exponents or exponents that are zero.
You cannot simplify a positive exponent attached to a variable.
Examples
NOT SIMPLIFIED
SIMPLIFIED
𝑥 −3
y3
𝑦0
m
3
𝑚−3 𝑛1
𝑥4𝑦2
Positive Exponents
Re-write each expression using exponents
x2  x3
1. x  x  x
2. x  x  y  y
3.
4. x 4  y 2  x 3  y
5. x  y 4  y
6. 5x 2 3x 4
  
Re-write each exponent as an expression without exponents.
7.
x6
8.
x2 y3
9.
m 2 ny 4
Product of Powers Rule
For any number a, and all integers m and n:
Examples:
10.
y3  y4 
11. (21c 6 )(c 7 ) 
a m  a n  a mn
12. (2a 4 )( 2a 3b 2 )( 3ab 3 ) 
How do you simplify exponents negative?
A negative exponent is notation for a number that is the reciprocal of that same number with a
positive exponent.
Reciprocal: Two numbers are called reciprocals of one another if their product is 1. To find the
reciprocal, make sure the number is in fraction form and “flip” it.
1
1
because 2   1
2
2
1
1
The reciprocal of , for example, is 4 because  4  1
4
4
7
8
7 8
The reciprocal of , for example, is
because   1
8
7
8 7
The reciprocal of 2, for example, is
5 2 
2 1

1 2
1 4
 4
4 1
7 8

8 7
2
Examples of Negative Exponents
1 32
1 2 1 32 32
2
2
2


3
4

3

3  2   2
32 1
42
4 1 4
1
52
1. The expression 𝑎−𝑛 is simplified: 𝑎−𝑛 = ________
2. The expression
1
𝑎−𝑛
is simplified:
1
= ________
𝑎−𝑛
𝑎≠0
𝑎≠0
Numerical Examples: Simplify each expression.
13.
8
17.
4  4 1
2
1
2 3
14.
18.
15.
(2 ∙ 3)−2
19.
4
7 2
32
16. 2
7
3−2 ∙ 32
20.
(4−2 )3
Algebraic Examples: Simplify each expression.
21
𝑥 −3
22.
𝑥 −1 𝑦 7 𝑚−5
25.
26.
𝑐 −5
23.
(2𝑥)−2
27.
1
𝑑 −4
𝑥
( )−2
𝑦
24.
𝑥 −2 𝑦 3
28.
2𝑥 −2 𝑦 −3
How do you simplify exponents that are zero?
Look at the following three examples and determine a pattern to fill out the
rule for exponents that are zero.
3
53  5  5  5
2  222
z3  z  z  z
23 2  2  2
53 5  5  5
z3 z  z  z
22  
 22
52  
 55
z2  
 zz
2
2
5
5
z
z
22 2  2
52 5  5
z2 z  z
1
1
1
2  
2
5  
5
z  
z
2
2
5
5
z
z
21 2
51 5
z1 z
20    1
50    1
z0    1
2 2
5 5
z z
1. The expression 𝑎0 is simplified: 𝑎0 = ________
𝑎≠0
Simplify each expression.
29.
90
30.
7
20
31.
x2 y0
32.
5 2  31  8 0