Download A.2 EXPONENTS AND RADICALS

A14
Appendix A
Review of Fundamental Concepts of Algebra
A.2
What you should learn
• Use properties of exponents.
• Use scientific notation to represent
real numbers.
• Use properties of radicals.
• Simplify and combine radicals.
• Rationalize denominators and
numerators.
• Use properties of rational exponents.
EXPONENTS AND RADICALS
Integer Exponents
Repeated multiplication can be written in exponential form.
Repeated Multiplication
a⭈a⭈a⭈a⭈a
Exponential Form
a5
共⫺4兲共⫺4兲共⫺4兲
共⫺4兲3
共2x兲共2x兲共2x兲共2x兲
共2x兲4
Why you should learn it
Exponential Notation
Real numbers and algebraic expressions
are often written with exponents and
radicals. For instance, in Exercise 121
on page A26, you will use an expression
involving rational exponents to find the
times required for a funnel to empty for
different water heights.
If a is a real number and n is a positive integer, then
an ⫽ a ⭈ a ⭈ a . . . a
n factors
where n is the exponent and a is the base. The expression an is read “a to the nth
power.”
An exponent can also be negative. In Property 3 below, be sure you see how to use a
negative exponent.
Properties of Exponents
T E C H N O LO G Y
You can use a calculator to
evaluate exponential expressions.
When doing so, it is important to
know when to use parentheses
because the calculator follows the
order of operations. For instance,
evaluate 共⫺2兲4 as follows.
Scientific:
冇
2
ⴙⲐⴚ
冈
yx
4
ⴝ
冇
冇ⴚ冈
2
冈
>
Graphing:
4
Let a and b be real numbers, variables, or algebraic expressions, and let m and n
be integers. (All denominators and bases are nonzero.)
Property
1. a a ⫽ a
m n
2.
2
3
am
⫽ am⫺n
an
3. a⫺n ⫽
⭈3
4
⫽ 32⫹4 ⫽ 36 ⫽ 729
x7
⫽ x7⫺ 4 ⫽ x 3
x4
冢冣
1
1
⫽
an
a
4. a0 ⫽ 1,
n
a⫽0
y⫺4 ⫽
冢冣
1
1
⫽
y4
y
4
共x 2 ⫹ 1兲0 ⫽ 1
5. 共ab兲m ⫽ am bm
共5x兲3 ⫽ 53x3 ⫽ 125x3
6. 共am兲n ⫽ amn
共 y3兲⫺4 ⫽ y3(⫺4) ⫽ y⫺12 ⫽
ENTER
The display will be 16. If you
omit the parentheses, the display
will be ⫺16.
Example
m⫹n
7.
冢b冣
a
m
⫽
am
bm
ⱍ ⱍ ⱍⱍ
8. a2 ⫽ a 2 ⫽ a2
冢x冣
2
3
⫽
1
y12
23
8
⫽
x3 x3
ⱍ共⫺2兲2ⱍ ⫽ ⱍ⫺2ⱍ2 ⫽ 共2兲2 ⫽ 4
Appendix A.2
Exponents and Radicals
A15
It is important to recognize the difference between expressions such as 共⫺2兲4 and
⫺2 . In 共⫺2兲4, the parentheses indicate that the exponent applies to the negative sign
as well as to the 2, but in ⫺24 ⫽ ⫺ 共24兲, the exponent applies only to the 2. So,
共⫺2兲4 ⫽ 16 and ⫺24 ⫽ ⫺16.
The properties of exponents listed on the preceding page apply to all integers m and
n, not just to positive integers, as shown in the examples in this section.
4
Example 1
Evaluating Exponential Expressions
a. 共⫺5兲2 ⫽ 共⫺5兲共⫺5兲 ⫽ 25
Negative sign is part of the base.
b. ⫺52 ⫽ ⫺ 共5兲共5兲 ⫽ ⫺25
Negative sign is not part of the base.
c. 2
⭈
24
⫽
21⫹4
⫽
25
⫽ 32
Property 1
4
d.
4
1
1
⫽ 44⫺6 ⫽ 4⫺2 ⫽ 2 ⫽
46
4
16
Properties 2 and 3
Now try Exercise 11.
Example 2
Evaluating Algebraic Expressions
Evaluate each algebraic expression when x ⫽ 3.
a. 5x⫺2
b.
1
共⫺x兲3
3
Solution
a. When x ⫽ 3, the expression 5x⫺2 has a value of
5x⫺2 ⫽ 5共3兲⫺2 ⫽
5
5
⫽ .
2
3
9
1
b. When x ⫽ 3, the expression 共⫺x兲3 has a value of
3
1
1
1
共⫺x兲3 ⫽ 共⫺3兲3 ⫽ 共⫺27兲 ⫽ ⫺9.
3
3
3
Now try Exercise 23.
Example 3
Using Properties of Exponents
Use the properties of exponents to simplify each expression.
a. 共⫺3ab4兲共4ab⫺3兲
b. 共2xy2兲3
c. 3a共⫺4a2兲0
Solution
a. 共⫺3ab4兲共4ab⫺3兲 ⫽ 共⫺3兲共4兲共a兲共a兲共b4兲共b⫺3兲 ⫽ ⫺12a 2b
b. 共2xy 2兲3 ⫽ 23共x兲3共 y 2兲3 ⫽ 8x3y6
c. 3a共⫺4a 2兲0 ⫽ 3a共1兲 ⫽ 3a, a ⫽ 0
5x 3 2 52共x 3兲2 25x 6
d.
冢y冣
⫽
y2
⫽
y2
Now try Exercise 31.
冢5xy 冣
3 2
d.
A16
Appendix A
Review of Fundamental Concepts of Algebra
Example 4
Rarely in algebra is there only
one way to solve a problem.
Don’t be concerned if the steps
you use to solve a problem are
not exactly the same as the steps
presented in this text. The
important thing is to use steps
that you understand and, of
course, steps that are justified
by the rules of algebra. For
instance, you might prefer the
following steps for Example 4(d).
冢 冣
3x 2
y
⫺2
冢 冣
y
⫽
3x 2
2
y2
⫽ 4
9x
Note how Property 3 is used
in the first step of this solution.
The fractional form of this
property is
冢冣
a
b
⫺m
Rewrite each expression with positive exponents.
a. x⫺1
b.
1
3x⫺2
c.
12a3b⫺4
4a⫺2b
2 ⫺2
d.
冢3xy 冣
Solution
1
x
a. x⫺1 ⫽
Property 3
b.
1
1共x 2兲 x 2
⫽
⫽
3x⫺2
3
3
The exponent ⫺2 does not apply to 3.
c.
12a3b⫺4 12a3 ⭈ a2
⫽
4a⫺2b
4b ⭈ b4
Property 3
d.
冢 冣
3x 2
y
⫺2
冢冣.
b
⫽
a
Rewriting with Positive Exponents
m
⫽
3a5
b5
Property 1
⫽
3⫺2共x 2兲⫺2
y⫺2
Properties 5 and 7
⫽
3⫺2x⫺4
y⫺2
Property 6
⫽
y2
32x 4
Property 3
⫽
y2
9x 4
Simplify.
Now try Exercise 41.
Scientific Notation
Exponents provide an efficient way of writing and computing with very large (or very
small) numbers. For instance, there are about 359 billion billion gallons of water on
Earth—that is, 359 followed by 18 zeros.
359,000,000,000,000,000,000
It is convenient to write such numbers in scientific notation. This notation has the form
± c ⫻ 10n, where 1 ≤ c < 10 and n is an integer. So, the number of gallons of water on
Earth can be written in scientific notation as
3.59
⫻
100,000,000,000,000,000,000 ⫽ 3.59 ⫻ 1020.
The positive exponent 20 indicates that the number is large (10 or more) and that
the decimal point has been moved 20 places. A negative exponent indicates that the
number is small (less than 1). For instance, the mass (in grams) of one electron is
approximately
9.0
⫻
10⫺28 ⫽ 0.0000000000000000000000000009.
28 decimal places
Appendix A.2
Example 5
Exponents and Radicals
A17
Scientific Notation
Write each number in scientific notation.
a. 0.0000782
b. 836,100,000
Solution
a. 0.0000782 ⫽ 7.82 ⫻ 10⫺5
b. 836,100,000 ⫽ 8.361 ⫻ 108
Now try Exercise 45.
Example 6
Decimal Notation
Write each number in decimal notation.
a. ⫺9.36 ⫻ 10⫺6
b. 1.345 ⫻ 102
Solution
a. ⫺9.36 ⫻ 10⫺6 ⫽ ⫺0.00000936
⫻
b. 1.345
102 ⫽ 134.5
Now try Exercise 55.
T E C H N O LO G Y
Most calculators automatically switch to scientific notation when they are showing
large (or small) numbers that exceed the display range.
To enter numbers in scientific notation, your calculator should have an exponential
entry key labeled
EE
or EXP .
Consult the user’s guide for your calculator for instructions on keystrokes and how
numbers in scientific notation are displayed.
Example 7
Evaluate
Using Scientific Notation
共2,400,000,000兲共0.0000045兲
.
共0.00003兲共1500兲
Solution
Begin by rewriting each number in scientific notation and simplifying.
共2,400,000,000兲共0.0000045兲 共2.4 ⫻ 109兲共4.5 ⫻ 10⫺6兲
⫽
共0.00003兲共1500兲
共3.0 ⫻ 10⫺5兲共1.5 ⫻ 103兲
⫽
共2.4兲共4.5兲共103兲
共4.5兲共10⫺2兲
⫽ 共2.4兲共105兲
⫽ 240,000
Now try Exercise 63(b).
A18
Appendix A
Review of Fundamental Concepts of Algebra
Radicals and Their Properties
A square root of a number is one of its two equal factors. For example, 5 is a square
root of 25 because 5 is one of the two equal factors of 25. In a similar way, a cube root
of a number is one of its three equal factors, as in 125 ⫽ 53.
Definition of nth Root of a Number
Let a and b be real numbers and let n ⱖ 2 be a positive integer. If
a ⫽ bn
then b is an nth root of a. If n ⫽ 2, the root is a square root. If n ⫽ 3, the root
is a cube root.
Some numbers have more than one nth root. For example, both 5 and ⫺5 are square
roots of 25. The principal square root of 25, written as 冪25, is the positive root, 5. The
principal nth root of a number is defined as follows.
Principal nth Root of a Number
Let a be a real number that has at least one nth root. The principal nth root of a
is the nth root that has the same sign as a. It is denoted by a radical symbol
n a.
冪
Principal nth root
The positive integer n is the index of the radical, and the number a is the radicand.
2
If n ⫽ 2, omit the index and write 冪a rather than 冪
a. (The plural of index is
indices.)
A common misunderstanding is that the square root sign implies both negative and
positive roots. This is not correct. The square root sign implies only a positive root.
When a negative root is needed, you must use the negative sign with the square root
sign.
Incorrect: 冪4 ⫽ ± 2
Example 8
Correct: ⫺ 冪4 ⫽ ⫺2
and 冪4 ⫽ 2
Evaluating Expressions Involving Radicals
a. 冪36 ⫽ 6 because 62 ⫽ 36.
b. ⫺ 冪36 ⫽ ⫺6 because ⫺ 共冪36兲 ⫽ ⫺ 共冪62兲 ⫽ ⫺ 共6兲 ⫽ ⫺6.
5 3 53 125
125 5
c. 3
⫽ because
⫽ 3⫽
.
64
4
4
4
64
5 ⫺32 ⫽ ⫺2 because 共⫺2兲5 ⫽ ⫺32.
d. 冪
4 ⫺81 is not a real number because there is no real number that can be raised to the
e. 冪
fourth power to produce ⫺81.
冪
冢冣
Now try Exercise 65.
Appendix A.2
Exponents and Radicals
A19
Here are some generalizations about the nth roots of real numbers.
Generalizations About nth Roots of Real Numbers
Real Number a
Integer n
Root(s) of a
Example
a > 0
n > 0, n is even.
n a,
n a
冪
⫺冪
4 81 ⫽ 3,
4 81 ⫽ ⫺3
冪
⫺冪
a > 0 or a < 0
n is odd.
n
冪
a
3
冪
⫺8 ⫽ ⫺2
a < 0
n is even.
No real roots
冪⫺4 is not a real number.
a⫽0
n is even or odd.
n 0 ⫽ 0
冪
5 0 ⫽ 0
冪
Integers such as 1, 4, 9, 16, 25, and 36 are called perfect squares because they
have integer square roots. Similarly, integers such as 1, 8, 27, 64, and 125 are called
perfect cubes because they have integer cube roots.
Properties of Radicals
Let a and b be real numbers, variables, or algebraic expressions such that the
indicated roots are real numbers, and let m and n be positive integers.
Property
1.
n am
冪
n a
2. 冪
3.
4.
n
冪
a
n b
冪
⫽共
兲
n a m
冪
3 82
冪
n b ⫽冪
n ab
⭈冪
冪5
⫽
冪ab ,
27
4 9
冪
m n
冪a ⫽ mn
冪a
冪
兲
⫽ 共2兲2 ⫽ 4
⫽
冪279 ⫽
4
4 3
冪
3 冪
6 10
冪
10 ⫽ 冪
n
5. 共冪
a兲 ⫽ a
共冪3 兲2 ⫽ 3
n
ⱍⱍ
ⱍ
n an
冪
ⱍ
冪共⫺12兲2 ⫽ ⫺12 ⫽ 12
n an ⫽ a .
6. For n even, 冪
For n odd,
⫽共
⭈ 冪7 ⫽ 冪5 ⭈ 7 ⫽ 冪35
4
冪
b⫽0
n
Example
3 8 2
冪
共⫺12兲 ⫽ ⫺12
⫽ a.
3
冪
3
ⱍⱍ
A common special case of Property 6 is 冪a2 ⫽ a .
Example 9
Using Properties of Radicals
Use the properties of radicals to simplify each expression.
a. 冪8
⭈ 冪2
3 5
b. 共冪
兲
3
Solution
a.
b.
c.
d.
冪8
共
⭈ 冪2 ⫽ 冪8 ⭈ 2 ⫽ 冪16 ⫽ 4
3 5
冪
兲3 ⫽ 5
3 x3 ⫽ x
冪
ⱍⱍ
6 y6 ⫽ y
冪
Now try Exercise 77.
3 x3
c. 冪
6 y6
d. 冪
A20
Appendix A
Review of Fundamental Concepts of Algebra
Simplifying Radicals
An expression involving radicals is in simplest form when the following conditions are
satisfied.
1. All possible factors have been removed from the radical.
2. All fractions have radical-free denominators (accomplished by a process called
rationalizing the denominator).
3. The index of the radical is reduced.
To simplify a radical, factor the radicand into factors whose exponents are
multiples of the index. The roots of these factors are written outside the radical, and the
“leftover” factors make up the new radicand.
WARNING / CAUTION
When you simplify a radical, it
is important that both expressions
are defined for the same values
of the variable. For instance,
in Example 10(b), 冪75x3 and
5x冪3x are both defined only
for nonnegative values of x.
Similarly, in Example 10(c),
4 共5x兲4 and 5 x are both
冪
defined for all real values of x.
ⱍⱍ
Example 10
Simplifying Even Roots
Perfect
4th power
Leftover
factor
4
4
a. 冪
48 ⫽ 冪
16
4 4
4
2 ⭈ 3 ⫽ 2冪
3
⭈3⫽冪
Perfect
square
Leftover
factor
b. 冪75x3 ⫽ 冪25x 2 ⭈ 3x
⫽ 冪共5x兲2 ⭈ 3x
⫽ 5x冪3x
c.
Find largest square factor.
Find root of perfect square.
共5x兲 ⫽ ⱍ5xⱍ ⫽ 5ⱍxⱍ
4
冪
4
Now try Exercise 79(a).
Example 11
Simplifying Odd Roots
Perfect
cube
Leftover
factor
3 24 ⫽ 冪
3 8
a. 冪
3 23
3 3
⭈3⫽冪
⭈ 3 ⫽ 2冪
Perfect
cube
Leftover
factor
3 24a4 ⫽ 冪
3 8a3
b. 冪
⭈ 3a
3
3
⫽ 冪共2a兲 ⭈ 3a
3 3a
⫽ 2a 冪
3
6
3
c. 冪⫺40x ⫽ 冪共⫺8x6兲 ⭈ 5
3 共⫺2x 2兲3
⫽冪
3
⫽ ⫺2x 2 冪
5
Find largest cube factor.
Find root of perfect cube.
Find largest cube factor.
⭈5
Find root of perfect cube.
Now try Exercise 79(b).
Appendix A.2
Exponents and Radicals
A21
Radical expressions can be combined (added or subtracted) if they are like
radicals—that is, if they have the same index and radicand. For instance, 冪2, 3冪2,
and 12冪2 are like radicals, but 冪3 and 冪2 are unlike radicals. To determine whether
two radicals can be combined, you should first simplify each radical.
Example 12
Combining Radicals
a. 2冪48 ⫺ 3冪27 ⫽ 2冪16
⭈ 3 ⫺ 3冪9 ⭈ 3
Find square factors.
Find square roots and
multiply by coefficients.
⫽ 8冪3 ⫺ 9冪3
⫽ 共8 ⫺ 9兲冪3
⫽ ⫺ 冪3
b.
3 16x
冪
⫺
3 54x 4
冪
Combine like terms.
Simplify.
⫽
⭈ 2x ⫺
3
3 2x
冪
⫽ 2 2x ⫺ 3x冪
3 2x
⫽ 共2 ⫺ 3x兲 冪
3 8
冪
3 27
冪
⭈
x3
⭈ 2x
Find cube factors.
Find cube roots.
Combine like terms.
Now try Exercise 87.
Rationalizing Denominators and Numerators
To rationalize a denominator or numerator of the form a ⫺ b冪m or a ⫹ b冪m,
multiply both numerator and denominator by a conjugate: a ⫹ b冪m and a ⫺ b冪m
are conjugates of each other. If a ⫽ 0, then the rationalizing factor for 冪m is itself,
冪m. For cube roots, choose a rationalizing factor that generates a perfect cube.
Example 13
Rationalizing Single-Term Denominators
Rationalize the denominator of each expression.
5
2
a.
b. 3
冪
冪
2 3
5
Solution
a.
b.
5
2冪3
⫽
5
2冪3
5冪3
⫽
2共3兲
5冪3
⫽
6
冪3
⭈冪
3 52
2
2 冪
⫽
⭈
3 2
3
3
冪
5 冪
5 冪
5
3 52
2冪
⫽ 3 3
冪5
3 25
2冪
⫽
5
3
冪3 is rationalizing factor.
Multiply.
Simplify.
3
冪
52 is rationalizing factor.
Multiply.
Simplify.
Now try Exercise 95.
A22
Appendix A
Review of Fundamental Concepts of Algebra
Example 14
Rationalizing a Denominator with Two Terms
2
2
⫽
3 ⫹ 冪7 3 ⫹ 冪7
Multiply numerator and
denominator by conjugate
of denominator.
3 ⫺ 冪7
⭈ 3 ⫺ 冪7
⫽
2共3 ⫺ 冪7 兲
3共3兲 ⫹ 3共⫺冪7 兲 ⫹ 冪7共3兲 ⫺ 共冪7 兲共冪7 兲
Use Distributive Property.
⫽
2共3 ⫺ 冪7 兲
共3兲2 ⫺ 共冪7 兲2
Simplify.
⫽
2共3 ⫺ 冪7 兲
9⫺7
Square terms of
denominator.
⫽
2共3 ⫺ 冪7 兲
⫽ 3 ⫺ 冪7
2
Simplify.
Now try Exercise 97.
Sometimes it is necessary to rationalize the numerator of an expression. For
instance, in Appendix A.4 you will use the technique shown in the next example to
rationalize the numerator of an expression from calculus.
WARNING / CAUTION
Do not confuse the expression
冪5 ⫹ 冪7 with the expression
冪5 ⫹ 7. In general, 冪x ⫹ y
does not equal 冪x ⫹ 冪y.
Similarly, 冪x 2 ⫹ y 2 does not
equal x ⫹ y.
Example 15
冪5 ⫺ 冪7
2
Rationalizing a Numerator
⫽
⫽
⫽
⫽
冪5 ⫺ 冪7
冪5 ⫹ 冪7
⭈ 冪5 ⫹ 冪7
2
共冪5 兲2 ⫺ 共冪7 兲2
2共冪5 ⫹ 冪7兲
Multiply numerator and
denominator by conjugate
of numerator.
Simplify.
5⫺7
Square terms of numerator.
2共冪5 ⫹ 冪7 兲
⫺2
⫺1
⫽
2共冪5 ⫹ 冪7 兲 冪5 ⫹ 冪7
Simplify.
Now try Exercise 101.
Rational Exponents
Definition of Rational Exponents
If a is a real number and n is a positive integer such that the principal nth root of
a exists, then a1兾n is defined as
n
a1兾n ⫽ 冪
a, where 1兾n is the rational exponent of a.
Moreover, if m is a positive integer that has no common factor with n, then
n a
a m兾n ⫽ 共a1兾n兲m ⫽ 共冪
兲
m
The symbol
and
n a m.
a m兾n ⫽ 共a m兲1兾n ⫽ 冪
indicates an example or exercise that highlights algebraic techniques specifically
used in calculus.
Appendix A.2
WARNING / CAUTION
Rational exponents can be
tricky, and you must remember
that the expression bm兾n is not
n b is a real
defined unless 冪
number. This restriction
produces some unusual-looking
results. For instance, the number
共⫺8兲1兾3 is defined because
3
冪
⫺8 ⫽ ⫺2, but the number
共⫺8兲2兾6 is undefined because
6 ⫺8 is not a real number.
冪
Exponents and Radicals
A23
The numerator of a rational exponent denotes the power to which the base is raised,
and the denominator denotes the index or the root to be taken.
Power
Index
n b
n bm
b m兾n ⫽ 共冪
兲 ⫽冪
m
When you are working with rational exponents, the properties of integer exponents
still apply. For instance, 21兾221兾3 ⫽ 2(1兾2)⫹(1兾3) ⫽ 25兾6.
Example 16
Changing From Radical to Exponential Form
a. 冪3 ⫽ 31兾2
2 共3xy兲5 ⫽ 共3xy兲5兾2
b. 冪共3xy兲5 ⫽ 冪
4 x3 ⫽ 共2x兲共x3兾4兲 ⫽ 2x1⫹(3兾4) ⫽ 2x7兾4
c. 2x 冪
Now try Exercise 103.
Example 17
T E C H N O LO G Y
Changing From Exponential to Radical Form
a. 共x 2 ⫹ y 2兲3兾2 ⫽ 共冪x 2 ⫹ y 2 兲 ⫽ 冪共x 2 ⫹ y 2兲3
3
>
There are four methods of evaluating radicals on most graphing
calculators. For square roots, you
can use the square root key 冪 .
For cube roots, you can use the
cube root key 冪3 . For other
roots, you can first convert the
radical to exponential form and
then use the exponential key ,
or you can use the xth root key
x
冪
(or menu choice). Consult
the user’s guide for your calculator
for specific keystrokes.
4 y3z
b. 2y3兾4z1兾4 ⫽ 2共 y3z兲1兾4 ⫽ 2 冪
c. a⫺3兾2 ⫽
1
1
⫽
a3兾2 冪a3
5 x
d. x 0.2 ⫽ x1兾5 ⫽ 冪
Now try Exercise 105.
Rational exponents are useful for evaluating roots of numbers on a calculator, for
reducing the index of a radical, and for simplifying expressions in calculus.
Example 18
Simplifying with Rational Exponents
5 ⫺32
a. 共⫺32兲⫺4兾5 ⫽ 共冪
兲
⫺4
⫽ 共⫺2兲⫺4 ⫽
1
1
⫽
共⫺2兲4 16
b. 共⫺5x5兾3兲共3x⫺3兾4兲 ⫽ ⫺15x(5兾3)⫺(3兾4) ⫽ ⫺15x11兾12,
9 a3 ⫽ a3兾9 ⫽ a1兾3 ⫽ 冪
3 a
c. 冪
d.
x⫽0
Reduce index.
3 冪
6
6
冪
125 ⫽ 冪
125 ⫽ 冪
共5兲3 ⫽ 53兾6 ⫽ 51兾2 ⫽ 冪5
e. 共2x ⫺ 1兲4兾3共2x ⫺ 1兲⫺1兾3 ⫽ 共2x ⫺ 1兲(4兾3)⫺(1兾3)
⫽ 2x ⫺ 1,
x⫽
1
2
Now try Exercise 115.
The expression in Example 18(e) is not defined when x ⫽
冢2 ⭈ 12 ⫺ 1冣
⫺1兾3
is not a real number.
⫽ 共0兲⫺1兾3
1
because
2
A24
Appendix A
A.2
Review of Fundamental Concepts of Algebra
EXERCISES
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
In the exponential form an, n is the ________ and a is the ________.
A convenient way of writing very large or very small numbers is called ________ ________.
One of the two equal factors of a number is called a __________ __________ of the number.
n
The ________ ________ ________ of a number a is the nth root that has the same sign as a, and is denoted by 冪
a.
n
冪
In the radical form a, the positive integer n is called the ________ of the radical and the number a is called the
________.
When an expression involving radicals has all possible factors removed, radical-free denominators, and a reduced index,
it is in ________ ________.
Radical expressions can be combined (added or subtracted) if they are ________ ________.
The expressions a ⫹ b冪m and a ⫺ b冪m are ________ of each other.
The process used to create a radical-free denominator is known as ________ the denominator.
In the expression bm兾n, m denotes the ________ to which the base is raised and n denotes the ________ or root to be taken.
SKILLS AND APPLICATIONS
In Exercises 11–18, evaluate each expression.
In Exercises 31–38, simplify each expression.
11. (a) 32 ⭈ 3
55
12. (a) 2
5
13. (a) 共33兲0
14. (a) 共23 ⭈ 32兲2
31. (a) 共⫺5z兲3
32. (a) 共3x兲2
15. (a)
3
3⫺4
4 ⭈ 3⫺2
2⫺2 ⭈ 3⫺1
17. (a) 2⫺1 ⫹ 3⫺1
18. (a) 3⫺1 ⫹ 2⫺2
16. (a)
(b) 3 ⭈ 33
32
(b) 4
3
(b) ⫺32
3
2
(b) 共⫺ 35 兲 共53 兲
(b) 48共⫺4)⫺3
(b) 共⫺2兲0
(b) 共2⫺1兲⫺2
(b) 共3⫺2兲2
20. 共8⫺4兲共103兲
22.
43
3⫺4
In Exercises 23–30, evaluate the expression for the given
value of x.
23.
25.
27.
29.
⫺3x 3, x ⫽ 2
6x 0, x ⫽ 10
2x 3, x ⫽ ⫺3
⫺20x2, x ⫽ ⫺ 12
34. (a) 共⫺z兲3共3z4兲
24.
26.
28.
30.
7x⫺2, x ⫽ 4
5共⫺x兲3, x ⫽ 3
⫺3x 4, x ⫽ ⫺2
12共⫺x兲3, x ⫽ ⫺ 13
(b)
25y8
10y4
12共x ⫹ y兲3
9共x ⫹ y兲
4 3 3 4
(b)
y y
7x 2
x3
r4
36. (a) 6
r
35. (a)
In Exercises 19 –22, use a calculator to evaluate the
expression. (If necessary, round your answer to three decimal
places.)
19. 共⫺4兲3共52兲
36
21. 3
7
33. (a) 6y 2共2y0兲2
(b) 5x4共x2兲
(b) 共4x 3兲0, x ⫽ 0
3x 5
(b) 3
x
(b)
37. (a) 关共x2y⫺2兲⫺1兴⫺1
38. (a) 共6x7兲0,
x⫽0
冢 冣冢 冣
a
b
(b) 冢
冣冢
b
a冣
⫺2
3
⫺2
(b) 共5x2z6兲3共5x2z6兲⫺3
In Exercises 39–44, rewrite each expression with positive
exponents and simplify.
39. (a) 共x ⫹ 5兲0, x ⫽ ⫺5
40. (a) 共2x5兲0, x ⫽ 0
(b) 共2x 2兲⫺2
(b) 共z ⫹ 2兲⫺3共z ⫹ 2兲⫺1
41. (a) 共⫺2x 2兲3共4x3兲⫺1
(b)
42. (a) 共4y⫺2兲共8y4兲
43. (a) 3n
44. (a)
⭈ 32n
⭈
3
x ⭈ xn
x2
xn
冢10冣
x y
(b) 冢
5 冣
a
b
(b) 冢
冣冢
b
a冣
a
a
(b) 冢
冣冢
b
b冣
x
⫺1
⫺3 4 ⫺3
⫺2
3
⫺2
⫺3
⫺3
3
Appendix A.2
In Exercises 45–52, write the number in scientific notation.
46. ⫺7,280,000
48. 0.00052
⫺0.000125
Land area of Earth: 57,300,000 square miles
Light year: 9,460,000,000,000 kilometers
Relative density of hydrogen: 0.0000899 gram per
cubic centimeter
52. One micron (millionth of a meter): 0.00003937 inch
45.
47.
49.
50.
51.
10,250.4
In Exercises 53– 60, write the number in decimal notation.
53. 1.25 ⫻ 105
54. ⫺1.801 ⫻ 105
55. ⫺2.718 ⫻ 10⫺3
56. 3.14 ⫻ 10⫺4
57. Interior temperature of the sun: 1.5 ⫻ 107 degrees
Celsius
58. Charge of an electron: 1.6022 ⫻ 10⫺19 coulomb
59. Width of a human hair: 9.0 ⫻ 10⫺5 meter
60. Gross domestic product of the United States in 2007:
1.3743021 ⫻ 1013 dollars (Source: U.S. Department
of Commerce)
In Exercises 61 and 62, evaluate each expression without
using a calculator.
61. (a) 共2.0 ⫻ 109兲共3.4 ⫻ 10⫺4兲
(b) 共1.2 ⫻ 107兲共5.0 ⫻ 10⫺3兲
2.5 ⫻ 10⫺3
(b)
5.0 ⫻ 102
6.0 ⫻ 108
62. (a)
3.0 ⫻ 10⫺3
In Exercises 63 and 64, use a calculator to evaluate each
expression. (Round your answer to three decimal places.)
冢
冣
0.11
365
67,000,000 ⫹ 93,000,000
(b)
0.0052
63. (a) 750 1 ⫹
64. (a) 共9.3
⫻
800
106兲3共6.1
⫻
10⫺4兲
(b)
共2.414 ⫻ 104兲6
共1.68 ⫻ 105兲5
In Exercises 65–70, evaluate each expression without using a
calculator.
65.
66.
67.
68.
(a)
(a)
(a)
(a)
271兾3
32⫺3兾5
100⫺3兾2
冢⫺ 64冣
125
70. (a) 冢⫺
27 冣
69. (a)
3 27
(b) 冪
8
(b) 363兾2
⫺3兾4
(b) 共16
81 兲
⫺1兾2
(b) 共94 兲
冪9
1
⫺1兾3
⫺1兾3
冢冪32冣
1
(b) ⫺ 冢
125 冣
(b)
1
A25
Exponents and Radicals
In Exercises 71–76, use a calculator to approximate the
number. (Round your answer to three decimal places.)
71. (a) 冪57
3 452
72. (a) 冪
73. (a) 共⫺12.4兲⫺1.8
74. (a)
7 ⫺ 共4.1兲⫺3.2
2
75. (a) 冪4.5 ⫻ 109
76. (a) 共2.65 ⫻ 10⫺4兲1兾3
5 ⫺273
(b) 冪
6
(b) 冪
125
⫺2.5
(b) 共5冪3兲
(b)
冢133冣
⫺3兾2
冢 23冣
⫺ ⫺
13兾3
3
(b) 冪
6.3 ⫻ 104
(b) 冪9 ⫻ 10⫺4
In Exercises 77 and 78, use the properties of radicals to
simplify each expression.
5 2 5
77. (a) 共冪
兲
78. (a) 冪12 ⭈ 冪3
5 96x5
(b) 冪
4 共3x2兲4
(b) 冪
In Exercises 79–90, simplify each radical expression.
79. (a) 冪20
3 16
80. (a) 冪
27
3 128
(b) 冪
(b) 冪75
4
81. (a) 冪72x3
(b)
冪18z
32a
(b) 冪
b
2
3
4
82. (a) 冪54xy4
83.
84.
85.
86.
87.
88.
(a)
(a)
(a)
(a)
(a)
(a)
(b)
89. (a)
3 16x5
冪
4 3x 4 y 2
冪
2冪50 ⫹ 12冪8
4冪27 ⫺ 冪75
5冪x ⫺ 3冪x
2
(b)
(b)
(b)
(b)
(b)
冪75x2y⫺4
5 160x 8z 4
冪
10冪32 ⫺ 6冪18
3 16 ⫹ 3冪
3 54
冪
⫺2冪9y ⫹ 10冪y
8冪49x ⫺ 14冪100x
⫺3冪48x 2 ⫹ 7冪 75x 2
3冪x ⫹ 1 ⫹ 10冪x ⫹ 1
(b) 7冪80x ⫺ 2冪125x
90. (a) ⫺ 冪x 3 ⫺ 7 ⫹ 5冪x 3 ⫺ 7
(b) 11冪245x 3 ⫺ 9冪45x 3
In Exercises 91–94, complete the statement with <, ⫽, or >.
冪113 䊏
冪3
91. 冪5 ⫹ 冪3 䊏冪5 ⫹ 3
92.
93. 5䊏冪32 ⫹ 22
94. 5䊏冪32 ⫹ 42
冪11
In Exercises 95–98, rationalize the denominator of the
expression. Then simplify your answer.
⫺2兾5
⫺4兾3
95.
97.
1
96.
冪3
5
冪14 ⫺ 2
98.
8
3
冪
2
3
冪5 ⫹ 冪6
A26
Appendix A
Review of Fundamental Concepts of Algebra
In Exercises 99 –102, rationalize the numerator of the
expression. Then simplify your answer.
99.
冪8
2
冪5 ⫹ 冪3
101.
3
100.
t ⫽ 0.03关125兾2 ⫺ 共12 ⫺ h兲5兾2兴, 0 ⱕ h ⱕ 12
冪2
3
冪7 ⫺ 3
102.
4
In Exercises 103 –110, fill in the missing form of the
expression.
Radical Form
103. 冪2.5
3 64
104. 冪
105.䊏
106.䊏
3 ⫺216
107. 冪
108.䊏
4 81 3
109. 共冪
兲
110.䊏
121. MATHEMATICAL MODELING A funnel is filled
with water to a height of h centimeters. The formula
Rational Exponent Form
䊏
䊏
811兾4
⫺ 共1441兾2兲
䊏
共⫺243兲1兾5
䊏
165兾4
represents the amount of time t (in seconds) that it will
take for the funnel to empty.
(a) Use the table feature of a graphing utility to find
the times required for the funnel to empty for
water heights of h ⫽ 0, h ⫽ 1, h ⫽ 2, . . . h ⫽ 12
centimeters.
(b) What value does t appear to be approaching as the
height of the water becomes closer and closer to
12 centimeters?
122. SPEED OF LIGHT The speed of light is approximately
11,180,000 miles per minute. The distance from the
sun to Earth is approximately 93,000,000 miles. Find
the time for light to travel from the sun to Earth.
EXPLORATION
TRUE OR FALSE? In Exercises 123 and 124, determine
whether the statement is true or false. Justify your answer.
In Exercises 111–114, perform the operations and simplify.
共2x2兲3兾2
111. 1兾2 4
2 x
x⫺3 ⭈ x1兾2
113. 3兾2 ⫺1
x ⭈x
123.
x 4兾3y 2兾3
112.
共xy兲1兾3
5⫺1兾2 ⭈ 5x5兾2
114.
共5x兲3兾2
In Exercises 115 and 116, reduce the index of each radical.
4 32
冪
115. (a)
6 x3
116. (a) 冪
(b) 共x ⫹ 1兲
4 共3x2兲4
(b) 冪
6
冪
4
In Exercises 117 and 118, write each expression as a single
radical. Then simplify your answer.
117. (a)
118. (a)
冪冪32
冪冪243共x ⫹ 1兲
(b)
(b)
4 2x
冪冪
3 10a7b
冪冪
119. PERIOD OF A PENDULUM The period T (in
seconds) of a pendulum is T ⫽ 2␲冪L兾32, where L is
the length of the pendulum (in feet). Find the period of
a pendulum whose length is 2 feet.
120. EROSION A stream of water moving at the rate of v feet
per second can carry particles of size 0.03冪v inches.
Find the size of the largest particle that can be carried
by a stream flowing at the rate of 34 foot per second.
The symbol
indicates an example or exercise that highlights
algebraic techniques specifically used in calculus.
The symbol
indicates an exercise or a part of an exercise in which
you are instructed to use a graphing utility.
x k⫹1
⫽ xk
x
124. 共a n兲 k ⫽ a n
k
125. Verify that a0 ⫽ 1, a ⫽ 0. (Hint: Use the property of
exponents am兾a n ⫽ am⫺n.)
126. Explain why each of the following pairs is not equal.
(a) 共3x兲⫺1 ⫽
3
x
(b) y 3
⭈ y2 ⫽ y6
(c) 共a 2b 3兲4 ⫽ a6b7
(d) 共a ⫹ b兲2 ⫽ a 2 ⫹ b2
(e) 冪4x 2 ⫽ 2x
(f) 冪2 ⫹ 冪3 ⫽ 冪5
127. THINK ABOUT IT Is 52.7 ⫻ 105 written in scientific
notation? Why or why not?
128. List all possible digits that occur in the units place
of the square of a positive integer. Use that list to determine whether 冪5233 is an integer.
129. THINK ABOUT IT Square the real number 5兾冪3
and note that the radical is eliminated from the
denominator. Is this equivalent to rationalizing the
denominator? Why or why not?
130. CAPSTONE
(a) Explain how to simplify the expression 共3x3 y⫺2兲⫺2.
(b) Is the expression
or why not?
冪x4 in simplest form? Why
3