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Precalculus
Fifth Edition
Mathematics for Calculus
James Stewart  Lothar Redlin

Saleem Watson
1.2
Exponents
and Radicals
Exponents and Radicals
In this section, we give meaning
to expressions such as am/n in which
the exponent m/n is a rational number.
• To do this, we need to recall some facts about
integer exponents, radicals, and nth roots.
Integer Exponents
Integer Exponents
A product of identical numbers is usually
written in exponential notation.
• For example, 5 · 5 · 5 is written as 53.
• In general, we have the following definition.
Exponential Notation
If a is any real number and n is a positive
integer, then the nth power of a is:
an = a · a · · · · · a
n factors
• The number a is called the base and
n is called the exponent.
E.g. 1—Exponential Notation
(a)
 
1 5
2
  21  21  21  21  21  
1
32
(b)  3    3    3    3    3   81
4
(c)  3    3  3  3  3   81
4
Rules for Working with Exponential Notation
We can state several useful
rules for working with exponential
notation.
Rule for Multiplication
To discover the rule for multiplication,
we multiply 54 by 52:
5  5   5  5  5  5  5  5    5  5  5  5  5  5 
4
2
4 factors
2 factors
6 factors
5 5
6
42
• It appears that, to multiply two powers
of the same base, we add their exponents.
Rule for Multiplication
In general, for any real number a and any
positive integers m and n, we have:
a a 
m
n
 a  a  ...  a  a  a  ...  a 
m factors
n factors
 a  a  a...  a = a m  n
m  n factors
• Therefore, aman = am+n.
Rule for Multiplication
We would like this rule to be true
even when m and n are 0 or negative
integers.
• For instance, we must have:
20 · 23 = 20+3 = 23
• However, this can happen only if 20 = 1.
Rule for Multiplication
• Likewise, we want to have:
54 · 5–4 = 54+(–4) = 54–4 = 50 = 1
• This will be true if 5–4 = 1/54.
• These observations lead to the following
definition.
Zero and Negative Exponents
If a ≠ 0 is any real number and n is
a positive integer, then
a0 = 1
and
a–n = 1/an
E.g. 2—Zero and Negative Exponents
(a) 

4 0
7
(b) x
1
1
1 1
 1
x
x
(c)  2 
3

1
 2 
3
1
1


8
8
Laws of Exponents
Familiarity with these rules is essential
for our work with exponents and bases.
• The bases a and b are real numbers.
• The exponents m and n are integers.
Law 3—Proof
If m and n are positive integers, we have:
 a    a  a  ...  a 
m
n
n
m factors
  a  a  ...  a  a  a  ...  a  ...  a  a  ...  a 
m factors
m factors
m factors
n group of factors
 a  a  ...  a  a mn
mn factors
• The cases for which m ≤ 0 or n ≤ 0 can be proved
using the definition of negative exponents.
Law 4—Proof
If n is a positive integer, we have:
 ab 
n
  ab  ab  ...  ab 
n factors
  a  a  ...  a    b  b  ...  b   a n b n
n factors
n factors
• We have used the Commutative and Associative
Properties repeatedly.
• If n ≤ 0, Law 4 can be proved using the definition
of negative exponents.
• You are asked to prove Laws 2 and 5 in Exercise 88.
E.g. 3—Using Laws of Exponents
(a) x 4 x 7  x 47  x 11
4
(b) y y
7
y
4 7
y
3
(Law 1)
1
 3
y
(Law 1)
9
c
9 5
4
(c) 5  c  c
c
(Law 2)
E.g. 3—Using Laws of Exponents
(d)  b
4

5
b
45
b
20
(e)  3 x   3 x  27 x
3
5
3
3
x
x
x
(f )    5 
2
32
2
5
(Law 3)
3
(Law 4)
5
(Law 5)
E.g. 4—Simplifying Expressions with Exponents
Simplify:
3
2
4 3
(a) (2a b )(3ab )
3
x y x
(b)   

y  z 
2
4
E.g. 4—Simplifying
Example (a)
 2a b  3ab 
  2a b  3 a (b ) 
  2a b  27a b 
(Law 3 )
  2  27  a a b b
( Group factors with same base)
 54a b
(Law 1)
3
2
4
3
2
3
2
3
14
3
3
3
6
3
3
4 3
12
2 12
(Law 4 )
E.g. 4—Simplifying
3
4
Example (b)
 
2
4
x
x y x
x y
  3
  
4
y
z
y
z
  

3
8 4
x y x
 3 4
y z
2
3
4
(Laws 5 and 4)
(Law 3)
y  1
 x x  3 4
y z
(Group factors with
x7y 5
 4
z
(Laws 1 and 2)

3
4

8
same base)
Simplifying Expressions with Exponents
When simplifying an expression, you will
find that many different methods will lead
to the same result.
• You should feel free to use any of the rules
of exponents to arrive at your own method.
Laws of Exponents
We now give two additional laws that
are useful in simplifying expressions with
negative exponents.
Law 7—Proof
Using the definition of negative exponents
and then Property 2 of fractions (Section 1-1),
we have:
n
n
m
m
a
1/ a
1 b
b

 n
 n
m
m
b
1/ b
a
1
a
• You are asked to prove Law 6
in Exercise 88.
E.g. 5—Simplifying Exprns. with Negative Exponents
Eliminate negative exponents and simplify
each expression.
4
6st
(a) 2 2
2s t
 y 
(b)  3 
 3z 
2
E.g. 5—Negative Exponents
Example (a)
We use Law 7, which allows us to move
a number raised to a power from
the numerator to the denominator (or vice
versa) by changing the sign of the exponent.
4
2
6st
6ss
 2 4
2 2
2s t
2t t
3
3s
 6
t
(Law 7)
(Law 1)
E.g. 5—Negative Exponents
Example (b)
We use Law 6, which allows us to change
the sign of the exponent of a fraction by
inverting the fraction.
 y 
 3z 3 


2
 3z 


 y 
9z 6
 2
y
3
2
(Law 6)
(Law 5s and 4)
Scientific Notation
Scientific Notation
Exponential notation is used by scientists as
a compact way of writing very large numbers
and very small numbers.
For example,
• The nearest star beyond the sun, Proxima Centauri,
is approximately 40,000,000,000,000 km away.
• The mass of a hydrogen atom is about
0.00000000000000000000000166 g.
Scientific Notation
Such numbers are difficult to read and
to write.
So, scientists usually express them
in scientific notation.
Scientific Notation
A positive number x is said to be written
in scientific notation if it is expressed as
follows:
x = a x 10n
where:
• 1 ≤ a ≤ 10.
• n is an integer.
Scientific Notation
For instance, when we state that the distance
to Proxima Centauri is 4 x 1013 km,
the positive exponent 13 indicates that
the decimal point should be moved 13 places
to the right:
4 x 1013 = 40,000,000,000,000
Scientific Notation
When we state that the mass of a hydrogen
atom is 1.66 x 10–24 g, the exponent –24
indicates that the decimal point should be
moved 24 places to the left:
1.66 x 10–24
= 0.00000000000000000000000166
E.g. 6—Writing Numbers in Scientific Notation
(a)327,900  3.279  10
5
5 places
(b)0.000627  6.27  10
4 places
4
Scientific Notation in Calculators
Scientific notation is often used on
a calculator to display a very large or
very small number.
• Suppose we use a calculator to square
the number 1,111,111.
Scientific Notation in Calculators
The display panel may show (depending
on the calculator model) the approximation
1.234568 12
or
1.23468
E12
• The final digits indicate the power of 10,
and we interpret the result as 1.234568 x 1012.
E.g. 7—Calculating with Scientific Notation
a ≈ 0.00046
b ≈ 1.697 x 1022
and
c ≈ 2.91 x 10–18
use a calculator to approximate
the quotient ab/c.
If
• We could enter the data using scientific notation,
or we could use laws of exponents as follows.
E.g. 7—Calculating with Scientific Notation

4

4.6  10
1.697  10
ab

18
c
2.91 10
4.6 1.697 

4  22 18

 10
2.91
36
 2.7  10
22

• We state the answer correct to two significant
figures because the least accurate of the given
numbers is stated to two significant figures.
Radicals
Radicals
We know what 2n means whenever n is
an integer.
To give meaning to a power, such as 24/5,
whose exponent is a rational number,
we need to discuss radicals.
Radicals
The symbol √ means:
“the positive square root of.”
Thus,
a  b means b  a and b  0
2
Radicals
Since a = b2 ≥ 0, the symbol a makes
sense only when a ≥ 0.
• For instance,
9 3
because 32  9 and 3  0
nth Root
Square roots are special cases of nth
roots.
The nth root of x is the number that,
when raised to the nth power, gives x.
nth Root—Definition
If n is any positive integer, then the principal
nth root of a is defined as follows:
n
a  b means b n  a
If n is even, we must have a ≥ 0 and b ≥ 0.
nth Roots
Thus,
4
3
81  3
8  2
because 3  81 and 3  0
4
because
 2 
3
 8
nth Roots
However, 8 , 4 8 , and 6 8 are not
defined.
• For instance, 8 is not defined because
the square of every real number is
nonnegative.
nth Roots
Notice that
4  16  4
2
but
 -4 
2
 16  4  4
• So, the equation a2  a is not always true.
• It is true only when a ≥ 0.
nth Roots
However, we can always write
a a
2
• This last equation is true not only for square roots,
but for any even root.
• This and other rules used in working with nth roots
are listed in the following box.
Properties of nth Roots
In each property, we assume that all
the given roots exist.
E.g. 8—Simplifying Expressions Involving nth Roots
Simplify these expressions.
3
(a) x
4
4
8
(b) 81x y
4
E.g. 8—Expressions with nth Roots Example (a)
3
x  x x
4
3
3
 x
3
3 3
x x
3
(Factor out the largest cube)
x
(Property 1)
(Property 4)
E.g. 8—Expressions with nth Roots Example (b)
4
81x y  81 x
8
4
4
4
 
 34 x
2
 3x y
2
8 4
4
y
y
4
(Property 1)
(Property 5)
(Property 5)
Combining Radicals
It is frequently useful to combine like
radicals in an expression such as
2 3 5 3
• This can be done using the Distributive Property.
• Thus,
2 3  5 3  2  5 3  7 3
• The next example further illustrates this process.
E.g. 9—Combining Radicals
Combine:
(a) 32  200
(b) 25b  b
3
if b  0
E.g. 9—Combining Radicals
Example (a)
32  200
 16  2  100  2
(Factor out largest squares)
 16 2  100 2
(Property 1)
 4 2  10 2  14 2
(Distributive Property)
E.g. 9—Combining Radicals
Example (b)
If b > 0, then
25b  b
3
 25 b  b
2
b
(Property 1)
5 b b b
(Property 5, b  0)
 5  b  b
(Distributive Property)
Rational Exponents
Rational Exponents
To define what is meant by a rational
exponent or, equivalently, a fractional
exponent such as a1/3, we need to use
radicals.
Rational Exponents
To give meaning to the symbol a1/n in a way
that is consistent with the Laws of Exponents,
we would have to have:
(a1/n)n = a(1/n)n = a1 = a
• So, by the definition of nth root,
1/ n
a
 a
n
• In general, we define rational exponents
as follows.
Rational Exponent—Definition
For any rational exponent m/n in lowest
terms, where m and n are integers and n > 0,
we define
a
m/n
 ( a ) or equivalently a
n
m
m/n
( a )
n
m
If n is even, we require that a ≥ 0.
• With this definition, it can be proved that the Laws
of Exponents also hold for rational exponents.
E.g. 10—Using the Definition of Rational Exponents
1/ 2
 4 2
2/3

(a) 4
(b) 8
 8
3
2
2 4
2
Alternate solution:
8
2/3

3
8 
2
3
64  4
E.g. 10—Using the Definition of Rational Exponents
(c) 125
(d)
1/ 3
1
3
x
4

1
1
1



1/ 3
3
125
125 5
1
x
4/3
x
4 / 3
E.g. 11—Using the Laws of Exponents
1/ 3
(a) a a
(b)
a
2/5
a
7/3
a
7/5
3/5
a
8/3
(Law 1)
a
a
2 / 5  7 / 5 3 / 5
6/5
(Laws 1and 2)
E.g. 11—Using the Laws of Exponents
(c)  2a b
3

4 3/2
2
3/2

a   b 
 2
3 3/2
3
 2 2a
a
3 3 / 2 
9/2
b
6
4 3/ 2
b
4 3 / 2 
(Law 4)
(Law 3)
E.g. 11—Using the Laws of Exponents
3/4
4
2
x
y



(d)  1/ 3   1/ 2 
 y
 x

2 x
3


3/4 3
y 
1/ 3 3
9/4
8x

y
 8x
11/ 4
y x
 y 4 x 1/ 2
y
3
4
1/ 2

 Laws 5, 4, and 7 
 Law 3 
 Laws 1 and 2 
E.g. 12—Writing Radicals as Rational Exponents


  2x
1/ 2
3
(a) 2 x 3 x

 3 x 
 6x
1/ 2 1/ 3
 6x
5/6
1/ 3
(Definition)
(Law 1)
E.g. 12—Writing Radicals as Rational Exponents
(b) x x
  xx
 x
x

1/ 2 1/ 2

3 / 2 1/ 2
3/4
(Definition)
(Law 1)
(Law 3)
Rationalizing the Denominator
Rationalizing the Denominator
It is often useful to eliminate the radical in
a denominator by multiplying both numerator
and denominator by an appropriate
expression.
• This procedure is called rationalizing
the denominator.
Rationalizing the Denominator
If the denominator is of the form a ,
we multiply numerator and denominator
by a .
• In doing this, we multiply the given quantity by 1.
• So, we do not change its value.
Rationalizing the Denominator
For instance,
1
1
1
a
a

1 


a
a
a
a a
• Note that the denominator in the last fraction
contains no radical.
Rationalizing the Denominator
In general, if the denominator is of the form
n
a m with m < n
then multiplying the numerator and
denominator by n a n  m will rationalize
the denominator.
• This is because (for a > 0)
n
a
m n
a
n m
 a
n
m  n m
 a a
n
n
E.g. 13—Rationalizing Denominators
2
2 3



3
3
3 3
(a)
(b)
(c)
2
1
3
7
x
2

3
3
1
3
x
2 3
x
x

1
1
1
7
7
2
a
a2
a2
3
3
7
3
x
x

3
x
x
7
a
5
7
a5


7 5
7 7
a
a
a
a
5