1. Integer Exponents
Powers

2. If a is a real number and n is a positive
integer,
n
a = a ⋅⋅a
⋅ 
a 
n factors
0
a =1
a
−n
1
= n
a
if a ≠ 0
if a ≠ 0

3. 3
4 = 4⋅4⋅4
0
6 =1
4
−3
1
= 3
4

4. Laws of Exponents
m n
a a =a
m+ n
(a )
m n
=a
mn
( ab) = a b
n
m
a
1
m− n
=a
= n − m if a ≠ 0
n
a
a
n
n
 a  = a if b ≠ 0
 
n
 b
b
 a
 
 b
−n
 b
= 
 a
n
if a ≠ 0, b ≠ 0
n n

5. 3 −2
x y
Write −1 4 so that all exponents are positive.
x y
3 −2
3
−2
x y
x y
= −1 ⋅ 4
−1 4
x y
x
y
=x
3− ( −1) − 2 − 4
4 −6
=x y
y
4
x
= 6
y

6. Simplify each expression. Express the
answer so only positive exponents occur.
 3x y 
 3 
 x y 
2
−2
=3
4
−2
= ( 3x
(x ) (y )
−1 −2
3 −2
2 −3
y
)
4 −1 −2
−2
= ( 3x y
=3 x y
2
−1
−6
)
3 −2
2
x
= 6
9y

7. From Decimal to Scientific Notation
1. Count the number N of places that the
decimal point must be moved in order to
arrive at a number x, where 1 < x < 10.
2. If the original number is greater than
or equal to 1, the scientific notation is
N
x × 10 . If the original number is
between 0 and 1, the scientific notation is
x × 10− N .

8. Write the number 5,100,000,000 in
scientific notation.
9
51 × 10
.

9. Write the number 0.00032 in scientific
notation.
3.2 × 10
−4

10. Write 4.3 × 10
−5
0.000043
as a decimal.