Download Chapter 3. Exponents

Chapter 3. Exponents
3.1. Integer Exponents
KYOTE Standards: CR 10; CA 1, CA 3
We looked at some arithmetic problems involving exponents in Chapter 1. We
introduced exponential notation, observing that the symbol 53 = 5⋅ 5⋅ 5 represents
the product of 3 factors of 5 multiplied together. We generalize this statement in
Definition 1.
Definition 1. If a is a real number and n is a positive integer, then a to the power
n is
a n = a ⋅ a ⋅... ⋅ a
(n factors of a)
The number a is called the base and the number n is called the exponent.
We did not discuss properties of exponents in Chapter 1, but we could have
discovered them. For example, notice that
53 ⋅ 54 = (5⋅ 5⋅ 5)(5⋅ 5⋅ 5⋅ 5) = 57
We generalize this result for any base a and positive integer exponents m and n to
discover the property a m ⋅ a n = a m+n . In words, if we multiply m factors of a by n
factors of a we obtain m + n factors of a .
We would like this property to be true even when m and n are 0 or negative
integers. For example, if this property is true, then we must have, for any a ≠ 0 ,
a 0 ⋅ a 3 = a 0+3 = a 3
If we divide by a 3 , we obtain a 0 = 1.
Similarly, if n is a positive integer and this property is true, then we must have, for
any a ≠ 0 ,
a n ⋅ a−n = a n−n = a 0 = 1
If we divide by a n , we obtain a −n =
1
1
. If we divide by a −n , we obtain a n = −n .
n
a
a
These observations lead to the following definition.
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Definition 2. If a ≠ 0 is any real number and n is a positive integer, then
a0 = 1
and
a −n =
1
an
an =
and
1
a−n
With this definition, we are able to state five properties of exponents that apply
whenever the base is any nonzero real number and the exponent is any integer,
positive, negative or zero. Some examples of how each property is applied are given.
Properties of Exponents
If a ≠ 0 and b ≠ 0 are real numbers, m and n are integers, then
Property
Examples
1. a m a n = a m+n
53 ⋅ 54 = 53+4 = 57
am
2. n = a m−n
a
37
= 37−2 = 35
2
3
3. (a m )n = a mn
(72 )3 = 72⋅3 = 76
4. (ab)n = a n bn
(3⋅ 5)2 = 32 ⋅ 52
 a  an
5.   = n
b b
 5  54
  = 4
 3 3
n
4
1
54
−8
3
1
= 3−8−(−2) = 3−6 = 6
−2
3
3
1
(73 )−2 = 73⋅(−2 ) = 7−6 = 6
7
1
1
1
(3⋅ 5)−2 = 3−2 ⋅ 5−2 = 2 ⋅ 2 = 2 2
3 5 3 ⋅5
1 4
−4
−4
 5
5
1 34 34
5
= ⋅ =
  = −4 =
1 4 54 1 54
3
3
3
52 ⋅ 5−6 = 52−6 = 5−4 =
Positive Integer Exponents
We can use the properties of exponents to simplify algebraic expressions involving
positive exponents as discussed in Example 1.
Example 1. Simplify the given expression, writing it without any negative
exponents.
2
3
 x3   x 4 y2 
x3
3 4 2
2
3 4 3
(a) 5
(b) (5u v ) (c) (3a b)(−2a b ) (d)   

x
 4y   z 
Solution. (a) We can apply Property 2 and Definition 2 to obtain
x3
= x 3−5 = x −2
5
x
1
= 2
x
am
Property 2: n = a m−n
a
1
Definition 2: a − n = n
a
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In this example, it is easier to use Property 1 and cancel to obtain
x3
x3
=
x 5 x3x 2
1
= 2
x
Property 1: a m a n = a m+n
Divide out (cancel) x 3
(b) We use Property 4 and Property 3 to obtain
(5u3v 4 )2 = 52 (u3 )2 (v 4 )2
= 25u6 v8
Property 4: (ab)n = a n bn
Property 3: (a m )n = a mn
(c) We use Property 4, Property 3 and Property 1 to obtain
(3a 2 b)(−2a 3b 4 )3 = (3a 2 b)(−2)3 (a3 )3 (b 4 )3
= (3a 2 b)(−8)a 9 b12
= −24a11b13
Property 4: (ab)n = a n bn
Property 3: (a m )n = a mn
Property 1: a m a n = a m+n
(d) We use all five properties to obtain
 x3 
 
 4y 
2
 x 4 y 2  (x 3 )2 (x 4 y 2 )3
⋅

 =
2
z3
 z  (4y)
x 6 (x 4 y 2 )3
= 2 2⋅
4 y
z3
x 6 (x 4 )3 (y 2 )3
= 2 2⋅
4 y
z3
x 6 x12 y 6
=
16y 2 z 3
x18 y 6
=
16y 2 z 3
x18 y 4
=
16z 3
3
 a  an
Property 5:   = n
b b
n
Property 3: (a m )n = a mn
Property 4: (ab)n = a n bn
Property 3: (a m )n = a mn
Property 1: a m a n = a m+n
Property 2:
am
= a m− n
n
a
Negative Integer Exponents
We can also use the properties of exponents to simplify algebraic expressions that
involve both positive and negative integer exponents. There is a strategy that can be
used to simplify this process and help reduce mistakes. It involves converting an
expression involving negative exponents to equivalent expression involving only
positive exponents. We explain this strategy in the following example.
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Example 2. Convert the expression
x −3 y−2
to an equivalent expression involving
x 5 y−7 z −4
only positive exponents.
Solution. We write the expression as the product of single factors, apply
Definition 2, and rewrite as a single expression with all positive exponents. We
obtain
x −3 y −2
1
1 1
= x −3 ⋅ 5 ⋅ y −2 ⋅ −7 ⋅ −4
5 −7 −4
x y z
x
y z
1 1 1
= 3 ⋅ 5 ⋅ 2 ⋅ y7 ⋅ z4
x x y
y7z4
= 3 5 2
xx y
Multiplication of fractions
Definition 2: a−n =
1 n 1
; a = −n
an
a
Multiplication of fractions
We do not include this detailed explanation in the examples that follow. Instead, we
simply write
y7z 4
x −3 y−2
=
x 5 y−7 z −4 x 3 x 5 y 2
Convert to positive exponents
Example 3. Simplify the given expression, writing it without any negative
exponents.
−2
−3
 a2b 
 5x −4 y3 
2 −3 2
−3 2 −5
(a) (−3x y ) ( x y )
(b)  3 
(c)  2 −1 
 c 
 2x y 
Solution. (a) We use Property 4, Property 3, Property 1 and convert to
positive exponents to obtain
(−3x 2 y−3 )2 (x −3 y 2 )−5 = (−3)2 (x 2 )2 (y −3 )2 (x −3 )−5 (y 2 )−5
= 9x 4 y−6 x15 y −10
= 9x19 y −16
9x19
= 16
y
Property 4: (ab)n = a n bn
Property 3: (a m )n = a mn
Property 1: a m a n = a m+n
Convert to positive exponent
(b) We use Property 5, Property 4, and convert to positive exponents to
obtain
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−2
 a2 b 
(a 2 b)−2
 3  = 3 −2
(c )
 c 
 a  an
Property 5:   = n
b b
n
a −4 b −2
= −6
c
c6
= 4 2
ab
Property 4: (ab)n = a n bn
Convert to positive exponents
(c) We use Property 5, Property 4, convert to positive exponents, and then
use Property 1 to obtain
−3
 5x −4 y 3 
(5x −4 y 3 )−3
 2 −1  =
(2x 2 y −1 )−3
 2x y 
 a  an
Property 5:   = n
b b
n
5−3 x12 y−9
= −3 −6 3
2 x y
23 x12 x 6
= 3 3 9
5 yy
8x18
=
125y12
Property 4: (ab)n = a n bn
Convert to positive exponents
Property 1: a m a n = a m+n
Example 4. Find the value of the algebraic expression
x −3 y
when x = −3 and
x −2 y−2
y = −2 .
Solution. It is a good idea to simplify this complicated expression before
evaluating it. We first convert to all positive exponents, simplify using
Property 1 and canceling to obtain
x −3 y
x2 y y2
=
x −2 y−2
x3
x 2 y3
= 3
x
x 2 y3
= 2
x x
y3
=
x
Convert to positive exponents
Property 1: a m a n = a m+n
Write x 3 = x 2 x
Divide out (cancel) x 2
It is now a simple matter to find the value of the simplified expression
and y = −2 to obtain
68
y3
at x = −3
x
(−2)3 −8 8
=
=
−3
−3 3
Exercise Set 3.1
Simplify the expression, writing it without any negative exponents.
1. (−2x 4 )3
2. (6y)3
3. (−3a 4 bc 5 )2
4. −3(−2 x 2 )3
5. (−7a 4 )2
6. a 2 (a 4 b3c)5
7.
( xy 2 z 3 )4
( x 3 y 2 z )3
(
8.
10. (−x y) z
9.
 −u 2 v3 
11.  4 
 4u v 
)
2 4 3
3
(2y3 )4
2y 5
(2a 3 )2 (3a 4 )
(a 3 )4
 2xy 6 z 
12.  2 3 
 xy z 
2
3
13. (2x 3 )−3
14. (ab 2 )−7
15. u3u−9
16. (−2x 5 )−2
17. (x 2 y−3 )−4
18. (−u−4 )3 (2u 5 )−2
19. (−4x 2 y −7 )3
20. −3x −2 y (2x 3 y−4 )−2
21. (−2a3b−4 c−7 )3
a−3b 4
22. −5 5
a b
 a −1 
23.  4 
 5b 
2
 −2a 4 
24.  2 
 b 
−2
 x 8   x −3 
27.  −2   
 x   2x 
(
−1
)
2
−3
 9y 
25.  −5 
y 
 x6 
26.  2 
 4x 
(−a 2 b3 )−2
28.
a −4 b2
 x −1 yz −2 
29.  −8 −5 
 x y z
−1
−2
 2a 2 b 
31. (3ab c)  3 
 c 
2
3
 x −4 
1
32. x 2 y −5  3 −2 
4
 y 
69
2
−3
 xy−2 z −3 
30.  2 3 −4 
x y z 
2
 x 5   6x −8 
33.  −3  

 2y   y 
3
2
Find the value of the algebraic expression at the specified values of its variable or
variables without using a calculator. Simplify the expression before evaluating. Check
your answer using a calculator.
34. x 3 x −4 ; x = 5
35. (x −2 )3; x = −2
36. (2x −1 )3; x = 3
37. (x 2 + y 2 )−1; x = −1, y = 2
38. x −2 + y−2 ; x = 1, y = 2
39. (x + y)−1; x = 3, y = 5
−2
−1
−1
40. x + y ; x = 3, y = 5
x
41.   ; x = −2, y = 3
 y
−2
x
42.   ; x = 1, y = 3
 y
43.
x −3
44. −2 ; x = 7
x
 x −2 
45. 
 ; x = 3, y = 7
 y 
x2
; x = −2
x −3
−1
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3.2. Square Roots
KYOTE Standards: CR 10; CA 1, CA 4
Definition 1. The square root of a nonnegative number a is that unique
nonnegative number, denoted by a , whose square is a . In other words, a is that
nonnegative number such that ( a )2 = a . The symbol
is called a radical sign.
The squares of positive integers, called perfect squares, all have integer square roots.
The number 9 is a perfect square since 9 = 32 , but 8 is not a perfect square since
there is no integer n such that 8 = n 2 . Perfect squares play an important role in
simplifying both square roots of positive integers and algebraic expressions
involving positive integers.
There are two rules we use to simplify square roots of positive integers and
algebraic expressions involving positive integers.
Product Rule for Square Roots
If a and b are positive real numbers, then
ab = a b .
Quotient Rule for Square Roots
If a and b are a positive real numbers, then
a
a
.
=
b
b
Example 1. Simplify the expression.
8
25
2
Solution. (a) Since 49 = 7 is a perfect square, 49 = 7
(b) Since 48 is not a perfect square, we look for a factor of 48 that is a
perfect square. Successive factoring gives
(a)
49
(b)
48
(c)
48 = 4 ⋅12 = 4 ⋅ 4 ⋅ 3 = 16 ⋅ 3
We then use the product rule for square roots to obtain
48 = 16 ⋅ 3 = 16 3 = 4 3
(c) We use the quotient rule and then the product rule to obtain
8
8
4⋅2
4 2 2 2
=
=
=
=
25
5
5
5
25
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We can use the product and quotient rules for square roots to simplify algebraic
expressions. The idea is to treat the variables involved in the same way as we treat
positive integers. For example, if x is a positive real number, then x 2 = x since x
is the unique positive number that when squared is x 2 . Similarly, we have
x 6 = (x 3 )2 = x 3 and
x18 = (x 9 )2 = x 9 . If the power of x is an odd integer, then
we use the product rule for square roots. For example, a simplified form of
x 9 is
x 9 = x8 x = x8 x = x 4 x
Example 2. Simplify the expression. Assume that all variables represent positive
numbers.
(a)
36x 4 y 7
(b)
27x 5
(c)
8x 2 y 4
x6y
Solution. (a) We simplify the expression
36x 4 y 7 = 36 x 4 y 7
(d)
40x 3 y 2 z
2x −5 y −3
36x 4 y 7 as follows:
Product Rule for Square Roots
36 = 6 ; Factor: y 7 = y 6 y
= 6 x 4 y6 y
= 6 x 4 y6 y
Product Rule for Square Roots
= 6x 2 y 3 y
Simplify:
(b) We simplify the expression
y6 = y3
x4 = x2 ,
36x 4 y 7 as follows:
27x 5 = 9 ⋅ 3x 4 x
Factor: 27 = 9 ⋅ 3, x 5 = x 4 x
= 9 x 4 3x
Product Rule for Square Roots
= 3x 2 3x
Simplify:
9 = 3,
(c) We first simplify the expression under the radical sign of
x4 = x2
8x 2 y 4
and
x6y
then simplify the resulting expression. We obtain
8x 2 y 4
8y 3
=
x6y
x4
=
=
Simplify:
8y 3
Quotient Rule for Square Roots
x4
4 ⋅ 2y 2 y
x
8x 2 y 4 8y3
= 4
x6 y
x
Factor: 8 = 4 ⋅ 2 , y3 = y 2 y
4
72
=
4 y 2 2y
x4
2y 2y
=
x2
Product Rule for Square Roots
Simplify:
4 = 2,
y2 = y ,
x4 = x2
40x 3 y 2 z
and
2x −5 y −3
(d) We first simplify the expression under the radical sign of
then simplify the resulting expression. We obtain
20x 3 y 2 z
40x 3 y 2 z
=
2x −5 y −3
x −5 y −3
Divide by 2
= 20x 3 x 5 y 2 y 3z
Convert to positive exponents
= 20x 8 y 5 z
Simplify: x 3 x 5 = x 8 , y 2 y 3 = y 5
= 4 ⋅ 5x 8 y 4 yz
Factor: 20 = 4 ⋅ 5 , y 5 = y 4 y
= 4 x 8 y 4 5yz
Product Rule for Square Roots
= 2x 4 y 2 5yz
Simplify:
4 = 2,
x8 = x 4 ,
y4 = y2
Exercise Set 3.2
Simplify the expression. Assume that all variables represent positive numbers.
1. 12x 6
2.
32x 2 y10
3.
81x 9
4.
25x 5 y 9
7.
8x 5 y17
8.
3x 8 y 5 z 7
5.
24z 7
6. 18y 3 z12
9.
20
x 20
10.
x6
y10 z 2
11.
32x 3
9y 2
12.
y 9 z13
w12
49
8x 5
15.
27y8
8z 3
16.
52y 8
45z12
x 5 yz 8
x 2 y −5
19.
9x −2 y 6
45x −7 y 2
20.
33x 2 y 6
9x −3
5 9
13.
72x y
14.
17.
18x 5
2x −2
18.
73
Find the value of the algebraic expression at the specified values of its variable or
variables in simplified form without a calculator. Simplify the expression before
evaluating. Check your answer using a calculator.
21.
x 3 y 4 ; x = 3, y = 2
22.
x 3 y 5 ; x = 4, y = 7
23.
44x 5 ; x = 7
24.
98x 6 ; x = 2
25.
18x 5
; x=4
2x −2
26.
x 3 y −4
; x = 3, y = 5
x 7 y −2
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3.3. Roots and Rational Exponents
KYOTE Standards: CA 1, CA 4
Roots
Square roots are special cases of nth roots. We separate the definition of nth roots
into two types, one for n an even positive integer such as 2 and the other for n an
odd positive integer such as 3. These two types of nth roots have slightly different
properties.
Definition 1. If n is an even positive integer, the nth root of a nonnegative real
number a is that unique nonnegative real number, denoted by n a , such that
( a)
n
n
= a . When n = 2 , we write
2
a as
a , the square root of a .
It follows that 4 81 = 3 since 34 = 81, and that 6 64 = 2 since 2 6 = 64 . On the other
hand, notice that −4 is not defined since there is no real number whose square is
−4 . This is true since the square of any real number is always positive or zero.
Similarly, 4 −81 is not defined since there is no real number whose fourth power is
−81. Thus when n is even, the nth root of a negative number is not defined for this
same reason.
Definition 2. If n is an odd positive integer greater than 1, the nth root of a real
number a is that unique real number, denoted by
n
a , such that
( )
n
a
n
= a.
The nth root of an odd integer n , unlike an even integer n , is defined for any real
number. We have, for example, 3 −8 = −2 since (−2)3 = −8 and 5 −32 = −2 since
(−2)5 = −32 .
Rational Exponents
To define what we mean by a rational, or fractional, exponent, we begin with
rational exponents of the form 1 n where n is a positive integer.
Definition 3. For any rational number 1 n , where n is a positive integer, we define
a1 n = n a
Here a is a real number and, if n is even, a ≥ 0 .
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We can then extend this definition to all rational exponents.
Definition 4. For any rational number m n in lowest terms, where m and n are
integers and n > 0 , we define
am n =
( a)
n
m
or equivalently a m n = n a m
Here a is a real number and, if n is even, a ≥ 0 .
The Properties of Exponents for integer exponents we considered in Section 3.1 also
hold for rational exponents.
Properties of Exponents
If p and q are rational numbers in lowest terms with positive integer
denominators, then
Property
1. a p a q = a p+q
ap
2. q = a p−q
a
( )
3. a p
q
= a pq
Examples
= 53 2+1 2 = 52 = 25
5 5
47 2
= 4 7 2−3 2 = 4 2 = 16
32
4
8−1 382 3 = 8−1 3+2 3 = 81 3 = 2
75 3
= 75 3−1 = 72 3 = 3 72
7
(2 )
(8)
32 12
34 4
= 23 4⋅4 = 2 3 = 8
4. ( ab) = a p b p
(16 ⋅ 3)
 a ap
5.   = p
b b
 16 
161 2 4
=
=
 
12
9
3
9
p
p
12
−2 3
12
)
1 3 −2
−2
= ( −2) =
(8⋅ 4)
= 81 3 ⋅ 41 3 = 2 3 4
8
 
 27 
=
13
=161 2 ⋅ 31 2 = 4 3
(
= (8)
23
1
4
82 3
(81 3 )2 4
=
=
272 3 (271 3 )2 9
Each property holds for real numbers a and b for which the expressions on both
sides of the equality are defined.
Note. The Product Rule for Square Roots can be written in exponential form
(ab)1 2 = a1 2 b1 2 as illustrated by the first example of Property 4. The Quotient Rule
a
for Square Roots can be written in exponential form  
b
the first example of Property 5.
12
=
a1 2
as illustrated by
b1 2
Rational exponents and their properties provide important ways to do calculations.
The square root of a number can be calculated using the square root key on any
calculator. However, if we want to compute the cube root or any other root of a
number, or any rational power of a number, we must use rational exponents. For
76
example, if we want to calculate the value of 3 2 , we must enter this number as 21 3
since there is no cube root key on a calculator.
The following examples show how roots, rational exponents and the properties of
exponents can be applied to evaluate numerical expressions and simplify algebraic
expressions.
Example 1. Write each expression in radical form as an expression in exponential
form and each expression in exponential form as an expression in radical form.
1
(a) 5 x 2
(b)
(c) 34 7
(d) 5−2 3
3
x
Solution. (a) We write
5
x2 = ( x2 )
15
= x2 5
Convert to exponential form
( )
Property 3: a p
Note: We could also use Definition 4 to write
5
q
= a pq
x 2 = x 2 5 directly.
(b) We write
1
x3
=
1
(x )
1
= 32
x
3 12
Convert to exponential form
( )
q
= a pq
( )
q
= a pq
Property 3: a p
(c) We write
34 7 = (34 )1 7
Property 3: a p
= 7 34
Convert to radical form
1
52 3
Property 2*:
(d) We write
5−2 3 =
=
=
1
(5 )
1
2 13
3
52
ap
= a p−q
q
a
( )
Property 3: a p
q
= a pq
Convert to radical form
77
*Note. The expression
5−2 3 =
ap
1
= a p−q in Property 2 gives q = a−q when p = 0 . Hence
q
a
a
1
.
52 3
Example 2. Find the value of the given expression.
23
 8
1
32
3
(a) 125
(b) 5
(c) 9
(d)  − 
 27 
32
Solution. (a) We write 3 125 in exponential form and use the properties of
exponents to evaluate it.
3
125 = (125)
13
= (5
)
3 13
Convert to exponential form
Factor: 125 = 53
( )
=5
Property 3: a p
(b) We write
5
q
= a pq
1
in exponential form and use the properties of exponents
32
to evaluate it.
1 1
= 
32  32 
15
5
=
=
=
Convert to exponential form
(1)1 5
(32 )
15
(1)1 5
(2 )
5
15
 a ap
Property 5:   = p
b b
p
Factor: 32 = 2 5
( )
1
2
Property 3: a p
q
= a pq
q
= a pq
(c) We use the properties of exponents to write
9 3 2 = ( 91 2 )
3
= 33
= 27
( )
Property 3: a p
Simplify: 91 2 = 9 = 3
Simplify: 33 = 27
( )
Note. We could also use Property 3 to write 9 3 2 = 9 3
12
= ( 729 ) = 27 but most of
us would need a calculator to find the square root of 729!
78
12
(d) We use the properties of exponents to write
 8
− 
 27 
23
( −8)
=
23
 a ap
Property 5:   = p
b b
p
272 3
((−8) )
=
( 27 )
13 2
( )
Property 3: a p
13 2
(−2)2
32
4
−
9
=
q
= a pq
Simplify: (−8)1 3 = 2 , 271 3 = 3
Simplify: (−2)2 = 4 , 32 = 9
Example 3. Simplify the given expression. Assume that all variables represent
positive numbers.
81x13
x18
5
(b)
(c)
(a) 32 x y
y 21
32
Solution. (a) We first write the expression in exponential form and use the
properties of exponents to obtain
3
3
6 17
3
32x 6 y17 = (32x 6 y17 )1 3
Convert to exponential form
= (32)1 3 (x 6 )1 3 (y17 )1 3
Property 4: ( ab) = a p b p
= (8⋅ 4)1 3 x 2 ( y15 y 2 )1 3
Factor: 32 = 8⋅ 4 , y17 = y15 y 2 , Property 3
= 81 341 3 x 2 (y15 )1 3 (y 2 )
p
13
Property 4: ( ab) = a p b p
p
= 2x 2 y 5 41 3 (y 2 )1 3
81 3 = 2 ; Property 3: ( a p ) = a pq
= 2x 2 y 5 (4y 2 )1 3
Property 4: ( ab) = a p b p
q
p
If we convert the answer to radical form, we obtain
3
32x 6 y17 = 2x 2 y 5 3 4y 2 .
(b) We first write the expression in exponential form and use the properties
of exponents to obtain
81x13  81x13 
=  21 
y 21
 y 
13
3
(81x13 )1 3
=
(y 21 )1 3
Convert to exponential form
 a ap
Property 5:   = p
b b
p
79
(81)1 3 (x13 )1 3
=
y7
(27⋅ 3)1 3 (x12 x)1 3
=
y7
3⋅ 31 3 x 4 x1 3
=
y7
3x 4 (3x)1 3
=
y7
( )
q
= a pq
( )
q
= a pq
Property 4: ( ab) = a p b p , Property 3: a p
p
Factor: 81 = 27⋅ 3 , x13 = x12 x
Property 4: ( ab) = a p b p , Property 3: a p
p
Property 4: ( ab) = a p b p
p
If we convert to radical form, we obtain
3
81x13 3x 4 3 3x
=
.
y 21
y7
(c) We first write the expression in exponential form and use the properties
of exponents to obtain
x18  x18 
= 
32  32 
15
5
Convert to exponential form
 a ap
Property 5:   = p
b b
p
(x18 )1 5
321 5
(x15 x 3 )1 5
=
2
3 35
xx
=
2
=
(32)1 5 = 2 ; Factor: x18 = x15 x 3
( )
Property 4: ( ab) = a p b p , Property 3: a p
p
If we convert to radical form, we obtain
5
x18 x 3 5 x 3
=
.
32
2
Exercise Set 3.3
Simplify the expression. Assume that all variables represent positive numbers.
1.
5.
3
8x11
2. 3 16x 7
3
48
y12
6.
3
64x 25
y8
3.
3
40x 6 y 5
4.
3
72x10 y 5
7.
3
54x18
y 27
8.
3
32x 28
27y15
80
q
= a pq
Write each radical expression as an exponential expression and each exponential
expression as a radical expression.
9.
1
3
10.
3
72
14. 7−5 2
13. x 2 3
53
11.
12.
1
x7
3
16. x −5 3
15. x 3 5
Evaluate each expression without a calculator. Check your answer using a calculator.
17.
3
8
27
21.
3
−
18.
54
2
3
−64
19.
5
32
20.
3
−27
3
8
23.
−3
5
96
24.
22.
25. 81 3
26. 641 4
1
16
4
5
3
27. (−32)1 5
3
5
40
28. 811 2
Find the value of the algebraic expression at the specified values of its variable or
variables without using a calculator. Check your answer using a calculator.
29. x 2 3; x = 8
30. x 3 4 ; x = 16
31. ( x 3 y)1 2 ; x = 2, y = 4
32. x 4 3; x = 8
 x4 
33.  2  ; x = 2, y = 3
y 
34. (x 2 − y)3 2 ; x = 5, y = −7
35. ( x 3 + y 3 )5 2 ; x = 1, y = 2
36. (xy 2 )−2 3; x = 2, y = −2
32
−5 2
2 −1 2
37. (x + y )
2
; x = 3, y = 4
 x2 
38.  2 
y 
81
; x = 3, y = 2