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Properties of Exponents Summary
a0
b0
Section 6.1
Using Properties of Exponents
a m a n  a mn
am
 a m n
an
Quotient Rule
a 1
0
a m 
a 
m n
 ab 
n
1
am
 a mn
 a nb n
n
Product Rule
a a
   n
b b
Zero Exponent
Negative-Exponent
Power to Power Rule
Power of a Product
n
Power of a Quotient
Simplifying Exponential Expressions
1. Remove all parentheses using power/multiplication rules
The Product Rule
2. Remove powers to powers using the power to power rule
3. Combine like bases so that each base appears only once
4. Rewrite all bases to the exponent zero as 1
5. Rewrite equation so that all exponents are positive
The fully simplified answer should only have positive
exponents and each base should only appear once
ac mbc n  abc m  n
Example
Simplify:
 7 x  9x    7  9  x
5
3
5x y  4x y   20x y
3
5
2
8
3
53
 63x8
a m a n  a mn
ac mbc n  abc m  n
The Quotient Rule
am
 a mn
n
a
a0
ax m ax

n
bx
b
mn
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ax m ax

bx n
b
Example
Simplify:
mn
ax m ax

bx n
b
Example
mn
Simplify:
 2 
3
 2 
7
  2 
8x4 y3
2 x
7 3
  2   16
4
48 x5 y 3 z 2
3 2 2

3x
y z
2
16 x y
 4 x 41 y 3  4 x3 y 3
Zero as an Exponent
a  1;
0
a0
a4
4 4
0

a

a
1
a4
The Negative-Exponent Rule
a0  1
Example
Simplify:
5
65 z 0 6 1 65
 5  5 1
65
6
6
x7 x 7 x 7  7   x 0  1
am 
Example
1
am
Simplify:
am 
1
am
(5)2 
1
1

2
(5)(5)
(5)
52 
1
1

2
(5)(5)
(5)
a0


1
25
1
25
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am 
Example
1
am
Simplify:
The Power Rule for Exponents
(Power to Power)
1
 (1)72  (1)49  49
2
7
3x 2 y 4 
a
4
3y
x2
a 
m n
Example
b 
5
Simplify:
4
b
 5  4 
Power of a Product
b
20
n n
ab

a
b
 
n
2
4
7
 a mn
 a mn
 32   (3)4  81
y 

m n
 y 28
 ab 
Example
n
 a nb n
Power of a Quotient
Simplify:
 3x 
4 3
 6x y 
3
12
 33 x 43  27x
2
6 2
 62 x3 2 y 2  36x y
n
an
a
   n
b
b
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n
Example
Simplify:
an
a
   n
b
b
2
 x 4  x 4 2
x8
 3   3 2  6
y
y
y 
Properties of Exponents Summary
a0
b0
a m a n  a mn
am
 a m n
an
Quotient Rule
a 1
0
a m 
a 
m n
3
 5x 2 
125
53 x 6 125x 6

 3   9 
x6 y9
y
y9
 y 
 ab 
n
1
am
 a mn
 a nb n
n
Product Rule
a a
   n
b b
Zero Exponent
Negative-Exponent
Power to Power Rule
Power of a Product
n
Power of a Quotient
Simplifying Exponential Expressions
1. Remove all parentheses using power/multiplication rules
2. Remove powers to powers using the power to power rule
3. Combine like bases so that each base appears only once
4. Rewrite all bases to the exponent zero as 1
5. Rewrite equation so that all exponents are positive
The fully simplified answer should only have positive
exponents and each base should only appear once
Example
8.3 105
Convert from Scientific Notation to
decimal notation.
 8.3 100000  830000
2.09 104  2.09  0.0001  0.000209
4.045 102  4.045  0.01  0.04045
Converting from Scientific to Decimal Notation
In a 10n change a number in scientific notation
to decimal notation by moving the decimal point in
a to the right if n is positive, and move the decimal
point to the left if n is negative, n number of places.
Example
Convert from decimal notation to
scientific notation.
56,000,000  5.6 107
6
4,090,000  4.09 10
6
0.0000067  6.7 10
6
0.0000036  3.6 10
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Example
Example
Perform the indicated operations and write your
answer in scientific notation.
In 2008, Congress proposed a 700 billion dollar
bailout ($7 x 1011) of the US banking system. If
there are 300 million people in the US, what will be
the cost per person in tax dollars if every person
paid an equal amount of this bailout cost?
 7.3 10 8.0 10 
6
4.8 107
3.0 103
8
 58.4 102  5.84 101
 1.6 1010
7 1011
3.0 108
 2.3 103
$2,333 per person
 2 103 with proper significant figures
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