Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
Exponent
An exponent tells how many times a number, called the base, is repeated in a product.
Example:
3  3  3  3  3  35 , where 3 is the base and 5 is the exponent.
Integer Exponents: For any nonzero number a,
a1  a; a 2  a  a; a3  a  a  a; a 4  a  a  a  a; etc.
When an exponent is not written, assume it is 1. For example: 4  41
Multiplying powers of the same base: Add the exponents
For any nonzero number a, and any integers m and n,
a m  a n  a m n
Example: 35  32  3  3  3  3  3  3  3
Using the commutative property: 35  32  3  3  3  3  3  3  3
Simplifying gives: 37
This is the same as if we had added the exponents: 35  32  352  37
Dividing powers of the same base: Subtract the exponents
For any nonzero number a, and any integers m and n,
am
 a m n
n
a
35 3  3  3  3  3 3  3  3  3  3


 3  3  3  33
2
3  3
3
3 3
35
This is the same as if we had subtracted the exponents: 2  352  33
3
Example:
Single bases inside parentheses with outside exponents:
To raise a power to another power, multiply the exponents.
For any nonzero number a, and any integers m and n,
( a m )n  a mn
 
Example: n 2  n  n   n  n   n  n   n  n   n  n   n  n   n10
5
5
 
5
This is the same as if we had multiplied the exponents: n 2  n 25  n10
Multiple bases inside parentheses with outside exponents:
Chapter 1, Section 1.6
Page 1
Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
For any nonzero numbers a and b, and any integer nm. This is the distributive property.
( ab)n  a nbn
Negative Exponents:
For any nonzero number a,
1
an  n
a
Exponents of 0:
For any nonzero number a,
a 0  1 , and
Fractional exponents:
For any nonzero number a,
1
3
1
2
a  a;
a  a;
a 
 a ;
2
2
2
2
2
2
a  a ;
1
4
2
3
a 
2
3
a 4a
3
 a ;
2
3
a  a ;
3
2
3
4
a 
 a
4
3
3
4
a  4 a3
Scientific Notation
Very large and very small numbers that include trailing or leading zeros are easier to read
when they are expressed in scientific notation.
For example:
32,000,000 is written as 3.2  107 in scientific notation and
0.00000408 is written as 4.08  10 6 in scientific notation
How to Convert a Number to Scientific Notation:
1. Position a decimal point between the first and second digits.
2. Count how many places you moved the decimal to the right or left, and that’s the
power of 10.
a. If you moved the decimal to the left, the power is positive (for large
numbers)
b. If you moved the decimal to the right, the power is negative (for small
numbers)
Examples
Convert 47, 000 to scientific notation.
Chapter 1, Section 1.6
Page 2
Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
One Solution
First, imagine the number as a decimal: 47, 000.
Next, move the decimal between the first two digits: 4.7000
Then count how many positions to the left you moved the decimal (four, in this case)
Write that as a power of 10: 4.7  10 4
Convert 0.007345 to scientific notation.
One Solution
Move the decimal between the first two digits: 7.345
Then count how many positions to the right you moved the decimal (three, in this case)
Write that as a power of 10: 7.345  10 3
Multiplying in Scientific Notation
To multiple two numbers written in scientific notation, multiply the coefficients, and then
add the exponents. This uses the commutative property.

 
Multiply: 1.4 102  2.0 105

One Solution
First multiply the coefficients: 1.4  2.0  2.8
Next, add the exponents of the powers of 10: 10 2  10 5  10 25   10 3
Finally, join your new coefficient to your new power of 10: 2.8  10 3
Another way to look at this question is to use the commutative property:
1.4 102  2.0 105  1.4 102  2.0 105
Rearranging gives: 1.4  2.0  102 105

 



Dividing in Scientific Notation
To divide two numbers written in scientific notation, divide the coefficients, and then
subtract the exponent of the denominator (the bottom number) from the exponent of the
numerator (the top number).


Divide: 3.6 103 / 1.8 104

One Solution
First divide the coefficients: 3.6 / 1.8  2.0
Chapter 1, Section 1.6
Page 3
Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
Next, subtract the exponents of the powers of 10: 10 3 / 10 4  10 34  10 7
Finally, join your new coefficient to your new power of 10: 2.0  10 7
Regents Problem
The expression 8 4  8 6 is equivalent to
(1) 824
(3) 82
(2) 8 2
(4) 810
One Solution
84  86  8 46  82
Sample Regents Problem
Two objects are 2.4  1020 centimeters apart. A message from one object travels to the
other at a rate of 12
.  105 centimeters per second. How many seconds does it take the
message to travel from one object to the other?
(1) 12
(3) 2.0  1015
.  1015
(2) 2.0  104
(4) 2.88  1025
One Solution
2.4 1020 2.4

 10 205  2 1015
5
1.2 10
1.2

Chapter 1, Section 1.6
Page 4

Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
REGENTS QUESTIONS
SOLUTIONS
060312a
(3)
1
The expression 3 2  33  3 4 is equivalent to
(1) 27 9
(3) 39
(2) 27 24
(4) 324
2
069911a
The expression 2 3  4 2 is equivalent to
(1) 2 7
(3) 85
(2) 212
(4) 86
(1)
2  42
3
2 3  (2 2 )
23  2 4
27
3
010008a
(2)
The expression ( x 2 z 3 )( xy 2 z) is equivalent to
(1) x 2 y 2 z 3
(3) x 3 y 3 z 4
(2) x 3 y 2 z 4
(4) x 4 y 2 z 5
4
010306a
(3)
The product of 3x 5 and 2x 4 is
(1) 5x 9
(3) 6x 9
(2) 5x 20
(4) 6x 20
5
080001a
(2)
The product of 2x 3 and 6x 5 is
(1) 10x 8
(3) 10x15
(2) 12x8
(4) 12x15
6
010205a
The product of 3x 2 y and -4xy3 is
(1) -12x 3 y4
(3) -12x 2 y3
(2) 12x 3 y 4
(4) 12x 2 y3
Chapter 1, Section 1.6
Page 5
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2
Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
7
(3)
089906a
The product of 4x 2 y and 2xy3 is
(1) 8x 2 y3
(3) 8x 3 y 4
(2) 8x 3 y3
(4) 8x 2 y4
8
(3)
080605a
What is the product of 10 x 4 y 2 and 3xy3 ?
(1) 30 x 4 y 5
(3) 30 x 5 y 5
(2) 30 x 4 y 6
9
(4) 30 x 5 y 6
(4)
060604a
What is the product of
1 2 3
x y
2
1
(2) x 3 y 4
9
(1)
10
1 2
1
x y and xy3 ?
3
6
1 2 3
x y
(3)
18
1 3 4
x y
(4)
18
(1)
010910a
The expression (6 x 3 y 6 ) 2 is equivalent to
(1) 36 x 6 y 12
(3) 12 x 6 y12
(2) 36 x 5 y 8
(4) 6 x 6 y 12
11
060813ia
What is half of 2 6 ?
(1) 13
(3) 23
(2) 16
(4) 25
12
080405a
When 9 x 5 is divided by 3x 3 , x  0 , the quotient
is
(1) 3x 2
(3) 27 x15
(2) 3x 2
(4) 27 x 8
Chapter 1, Section 1.6
Page 6
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6
2
 25
1
2
(2)
Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
13
(3)
060005a
The quotient of 
(1) -3x 4
(2) -10x 4
14
15 x 8
, x  0, is
5x 2
(3) -3x 6
(4) -10x 6
(4)
060707a
The expression
(1) 8 x 4
(2) 8 x 6
15
32 x 8
, x  0, is equivalent to
4x2
(3) 8 x 4
(4) 8 x 6
(2)
080526a
5x 6 y 2
The expression
is equivalent to
x8 y
(1) 5 x 2 y
(2)
5y
x2
16
(3) 5 x14 y 3
(4)
5y3
x 14
(1)
010817a
2x
y
4x2 y3
The expression
is equivalent to
2 xy 4
2x
(1)
(3) 2xy
y
2y
(2)
(4) 2xy
x
17
(1)
060903ia
Which equation represents
form?
(1) 3x12 y 4
(2) 3x3 y5
Chapter 1, Section 1.6
Page 7
27 x18 y 5
9 x6 y
(3) 18x12 y 4
(4) 18x3 y5
in simplest
Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
18
(3)
fall0703ia
Which
expression
represents
simplest form?
(1) x 2
(2) x 9
19
3
(2 x 3 )(8 x 5 )
4x6
in
27 k 5m8
(4 k 3 )(9m2 )
1
(2 x )(8 x ) 16 x 8

 4x2
4x6
4x6
(3) 4 x 2
(4) 4 x 9
010932ia
Simplify:
5
060927ia
3k 2 m6
4
(4)
What is the product of 12 and 4.2  106 expressed in
scientific notation?
(1) 50.4  106
(3) 5.04  106
(2) 50.4  107
(4) 5.04  107
2
010927ia
(4)
What is the product of 8.4  108 and 4.2  103 written
in scientific notation?
(1) 2.0  105
(3) 35.28  1011
(2) 12.6  1011
(4) 3528
.
 1012
3
060207a
(1)
If 3.85 106 is divided by 385 104 , the result is
(1) 1
(3) 3.85 102
(2) 0.01
(4) 3.85 1010
4
010319a
6.3 108
What is the value of
in scientific notation?
3 104
(1) 2.1 10-2
(3) 2.1 10-4
(2) 2.1102
(4) 2.1104
Chapter 1, Section 1.6
Page 8
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Chapter 1: Sets, Operations and Algebraic Language
1.6: Exponents and Scientific Notation
5
fall0725ia
(2)
What is the quotient of 8.05  106 and 35
.  102 ?
(1) 2.3  103
(3) 2.3  108
(2) 2.3  104
(4) 2.3  1012
6
010018a
(4)
If the number of molecules in 1 mole of a substance is
6.02  1023 , then the number of molecules in 100 moles
is
(1) 6.02  1021
(3) 6.02  1024
(2) 6.02  1022
(4) 6.02  1025
7
060429a
(4)
If the mass of a proton is 167
.  1024 gram, what is the
mass of 1,000 protons?
(1) 1.67 10-27 g
(3) 1.67 10-22g
(2) 1.67 10-23g
(4) 1.67 10-21g
8
060815b
(4)
In 1995, the federal government paid off one-third of
its debt. If the original amount of the debt was
$4,920,000,000,000, which expression represents the
amount that was not paid off?
(1) 164
(3) 3.28  108
.  104
(2) 164
(4) 328
.  1012
.  1012
9
060628a
What is the sum of 6  10 3 and 3  10 2 ?
(1) 6.3  10 3
(3) 9  10 6
(2) 9  10 5
(4) 18  10 5
Chapter 1, Section 1.6
Page 9
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