College Mathematics Notes
Section 1.5
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Chapter 1: Pre-Algebra
Section 1.5: Exponents
Big Idea for this section: Exponents are a shortcut for writing repeated multiplication. There are rules for
simplifying calculations with many occurrences of exponents.
Big Skills for this section: You should be able to simplify expressions with exponents and use scientific
notation.
Section 1.5.1 Definition of Positive Integer Exponents
Exponents were originally developed as a shortcut notation for repeated multiplication:
Section 1.5.2 Simplifying Expressions of the Form am an
College Mathematics Notes
Section 1.5
Section 1.5.3 Simplifying Expressions of the Form (ab)n
Section 1.5.4 Simplifying Expressions of the Form (am)n
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College Mathematics Notes
Section 1.5
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Section 1.5.5 Simplifying Expressions Involving Two or More of These Forms
Some expressions will involve more than one of the exponent laws. You will need to carefully determine which
laws to apply. In general, start by simplifying within any parentheses groups. Work your way outward, applying
exponents to the factors of each group, and then finish by multiplying the simplified groups together.
Example
Simplify (3x3y4)2.
Example
Simplify (a2b)3(2a4).
Example
Simplify (5p4q3)2(3pq)2.
Rules of Exponents
x n  x m  x n m
 x  y n  x n y n
 
xm
n
 x mn
College Mathematics Notes
Section 1.5
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Section 1.5.6 The Meaning of Zero and Negative Exponents
If we look at the pattern of repeatedly dividing a number that is a perfect power by the base, we can see a
meaningful value for a base raised to a power of zero, and for a base raised to a negative power:
16  24
8  23
4  22
2  21
1  20
1
 21
2
1
 22
4
1
 23
8
1
 24
16
Any base (except a base of zero) raised to a power of zero is equal to one. Here is another explanation:
x 3  1xxx (1 times 3 factors of x)
x 2  1xx (1 times 2 factors of x)
x1  1x (1 times 1 factors of x)
x 0  1 (1 times no factors of x)
Any base raised to a negative power is the same as that base raised to the positive of that power, but on the other
side of the fraction bar.
Section 1.5.7 Simplifying Exponential Expressions Using the Property
Example
Simplify
am
 a mn
an
x5
.
x3
So to divide expressions with the same variable base, we subtract the exponents. The quotient property can be
applied to both numeric and variable expressions, as long as the numerator and denominator have powers of the
same base.
Examples
27
 2 7  4  23  8
4
2
w10
 w101  w9
w
College Mathematics Notes
Section 1.5
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Section 1.5.8 Simplifying Exponential Expressions With an exponent of Zero
Examples
w0 =
260 =
(5b)0 =
Note that the 0 power applies to the whole group.
5b0 =
Note that the 0 power applies only to what comes in front of it; namely, the “b”.
(3x)0 – 10x0 =
Section 1.5.9 Simplifying Exponential Expressions With a Negative Exponent
 Write as a fraction with a positive exponent in the denominator
Example
Simplify
x3
.
x8
Example
Simplify
c3
.
c5
Examples
32 
8 1 
w 4 
(4 y)2 
x 3  x 7 
p 5

p 2
College Mathematics Notes
Section 1.5
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Rules of Exponents
x n  x m  x n m
n
x
xn

 
yn
 y
xn
 x n m
m
x
 
xm
n
 x mn
 x  y n  x n y n
x n 
1
xn
Section 1.5.10 Scientific Notation
Exponents can be used to write very large and very small numbers in a more concise way, called scientific
notation.
Scientific notation is a way to write either a very large number or a number very close to zero using a number
between 1 and 10 times a power of ten.
Scientific notation is considered simpler because there is no need to write a long string of 0 digits at the end of a
large number, or long string of 0’s in a decimal that is very close to zero.
Writing Numbers in Scientific Notation
 Write the number as a number between 1 and 10 times a power of ten:
148 =
1500 =
3,780,000,000 =
0.051 =
0.000 568 =
0.000 000 0910 =
 shortcut: count how many times you have to move the decimal to get it behind the first significant digit; that
count is the positive power of ten for large numbers or the negative power of ten for numbers close to zero
45,000,000 =
7,810,000,000 =
540 =
College Mathematics Notes
Section 1.5
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0.0000003 =
0.000764 =
Change numbers written in scientific notation to standard (decimal) form
Do the multiplication longhand…
1.52  103 =
3.67  105 =
1.99  1012 =
9 104 
2.3 106 =
 shortcut: move the decimal left or right by the power of ten. Positive exponents make a big number,
negative exponents make a decimal close to zero.
Section 1.5.11 Multiplying and Dividing Numbers Using Scientific Notation
Using scientific notation can simplify calculations involving large numbers. First, convert numbers to scientific
notation, and then use exponent rules to reduce the powers of 10.
Example
Simplify (320,000)(50,000,000).