IMC
Unit 2: Exponents and Radicals
Properties of Exponents
5x
8
x 5 means
Properties of Exponents
o Each new operation is related to a previously taught operation
(each operation is a “step down”)
 Adding and Subtraction – basic (do not change exponent)
 Multiplication steps down to addition of exponents
 Division steps down to subtraction of exponents
 Exponents (power to a power) steps down to multiplication of exponents
Power
Multiplication
Division
Addition
Subtraction
o Move negative exponents before combining and simplifying (but follow order of
operations)
Multiplication Properties of Exponents
Let’s see what happens when we write out this problem…
4
3
a a 
This suggests our first rule for multiplying powers:
am  an 
43  4 2 
*If you multiply powers with the SAME base, then you
ADD the exponents!
NEVER touch the bases when dealing with the
exponents!!!!!!!
Ex. 1)
45  43
Ex. 2)
2 y 4 y 
3
Ex. 3)
x2  x  x7
Ex. 4)
(3)(3)2
Let’s see what happens when we write out this problem…
x 
2 3

This suggests our second rule for multiplying powers:
a 
4 
3 2

m n
5 
2 3
Ex. 5)
Ex. 6)
x 
7 3
Ex. 7)

x  2 x y 
4 3
7
Let’s see what happens when we write out this problem…
5x 
3

This suggests our third rule for multiplying powers:
a  b 
m
Ex. 8)
3x y 
4
3x 

2
Ex. 9)
4
 2a 
5 3
Ex. 10)

5x  3x
4 2
Ex. 11) 5xy  
2 3
Negative and Zero Exponents
Our only rule for zero exponents:
a 
6 
0
0
Let’s see what happens when we write out this problem…
a3

7
a
This suggests our only rule for negative exponents:
a
n

2
5 
Ex. 12)
3
 
4
3
5
2
Ex. 13)
 3x 2 y 3
Ex. 14)
4a  6
Ex. 16a)
Ex. 18)
50
Ex. 16b)
2 2 2
0
7x 3 y 2
Ex. 15) 14 a 8
3
6
00
Ex. 19)
Ex. 17)
(32 ) 3
x0  y0
5a 
2 0
Ex. 20)
5a 
Ex. 21)
4 2
Ex. 22)
4
 
Ex. 24)  3 
Ex. 26)
2
83  83
3x y 
3
Ex. 23)
2
 
3
5 2
1
 3x 
 1 
Ex. 25)  2 x 
2
Ex. 27)
1
27  23
Division Properties of Exponents
Let’s see what happens when we write out this problem…
5
a

2
a
This suggests our first rule for dividing powers:
m
a

n
a
6
5

3
5
Here is our second rule for dividing powers:
m
a
  
b
3
2
  
5
47
Ex 28) 2
4
x9
Ex 29) 3
x
a3
Ex 30) 5
a
3
Ex 31) 2
3
1
4

Ex 32)
42
63  6 2
Ex 33)
66
12n 2 m 5
Ex 34)
3n 4 m
 24a 6 b 7
Ex 35)
16b 4
x8 y 2
Ex 36) 3 5
x y
1
Ex 37)  
 4
3
 2x 
Ex 40)  y 2 z 3 


4
3x 3 y 12 x 2 y 2

Ex 43)
4x
y3
3
x 
x 
 3
Ex 38)  
 2
3 4
4b 
3 2
5 2
Ex 41)
4
Ex 42)
4a b 
2
2 3
 3x 2 y xy 2
Ex 45) 2 x 1  9 y 4
3
Ex 39)  
 y
3
  9 x3 y 2 

Ex 43) 
4
 3xy 
2